The January signing, making just his second appearance for the club, helped deliver the Blues' first victory of 2018 and inflict Leicester's first defeat of the calendar year with a fine first-half double.
The winger first tapped in from Gylfi Sigurdsson's cross before volleying in from Michael Keane's headed knockdown, both times after Everton won possession high up the pitch.
Jamie Vardy's 71st-minute penalty, coming after Wayne Rooney tripped Wilfried Ndidi, offered Leicester hope of rescuing a result.
But Kelechi Iheanacho was denied by the woodwork twice in quick succession while Matty James saw a shot cleared from the line as the Foxes fell to a first defeat in seven.
Walcott nets twice to put hosts in charge
Goalscorer Walcott forced the game's first save early on when Kasper Schmeichel had to dive low to push his 20-yard attempt wide, before Ndidi crashed a header off the crossbar from Marc Albrighton's cross at the other end.
Everton lacked any cohesion between their midfield and attack, as evidenced by their frequently sloppy passing, which frustrated the Goodison crowd as Leicester appeared the brighter side.
But the opener came from the kind of tenacity and aggression that the home side had lacked on 25 minutes.
Full-back Cuco Martina robbed Ben Chilwell of possession on the corner of the Leicester box to allow Sigurdsson to break into space and roll a ball across the area for an unmarked Walcott to steer home.
Everton claimed control after that and Oumar Niasse wasted a golden opportunity to make it 2-0 after Daniel Amartey could only clear Sigurdsson's cross as far as the striker, but he sliced wide with the goal at his mercy.
Yet Everton had a crucial second with 39 minutes gone with an accomplished first-time volley from Walcott after Keane headed Rooney's lofted ball across the box for the winger to drive beneath Schmeichel at the near post.
Niasse had another moment to forget when he almost ran clear through on goal immediately from kick off after the second goal, only to take a heavy first touch which allowed Schmeichel to gather.
Leicester's response after the break was positive although chances were sparse and Sigurdsson went close to putting the game beyond the away side after Walcott's cross, Schmeichel producing a good close-range save.
Everton soon resumed command and should have ended the match as a contest when Walcott's cross evaded Sigurdsson and Niasse, while Coleman almost set up Niasse for a tap in with another delivery from the right.
Vardy ramps up the tensions
But the hosts offered Leicester a route back into the game when Rooney brought down Ndidi in the box and Vardy stepped up from 12 yards to smash past Pickford for his 12th of the campaign to reduce the lead to a single goal.
Leicester stepped up the pressure with substitute Iheanacho hitting the crossbar and the post within seconds of each other, Jordan Pickford doing well to get a hand to both efforts.
Martina then denied midfielder James on the line after the midfielder broke into the box late to meet Chilwell's cross, with the rising tensions tangible.
Sam Allardyce introduced Morgan Schneiderlin and Dominic Calvert-Lewin as Everton looked to quell Leicester's growing threat and Aleksandar Dragović had to clear off the line from Walcott as they almost avoided a nervy finale.
Everton had another chance to add a third right at the death but Coleman ran out of steam and under-hit his pass in search of Calvert-Lewin on the counter-attack.
But Everton held on for a much-needed victory which keeps them three points ahead of 10th-placed Bournemouth, who thrashed Chelsea. Leicester drop down to eighth, behind Burnley.
Walcott gets Blues back in business
A seven-game winless streak had sparked concerns that Everton would be dragged back into a battle at the wrong end of the table, particularly given their struggles in the final third.
The Blues went 251 minutes between shots on target in the Premier League, a barren run spread across four games. Their lack of creativity was best demonstrated by £27 million January addition Cenk Tosun's failure to hit the target in either of his first two games after the excitement at his arrival. He was dropped here.
But such concerns were cast aside here with Everton much better going forward, largely through the duo of goalscorer Walcott and the returning Séamus Coleman.
Without truly blowing Leicester away, Everton were deserving winners and had enough chances to win more comfortably than they did - giving away a needless penalty to invite the Foxes back into the game at a time when they were in control.
Oumar Niasse - typically hard-working but short on finesse and finishing quality - was guilty of wasting some fine opportunities, while a few dangerous crosses went wasted.
It was former Arsenal winger Walcott, signed for £20 million earlier this month, who applied the cutting edge and quality in the final third that Everton have so sorely missed of late.
He followed up an assist on his debut with a first brace since October 2016 amidst a performance brimming with pace, purpose, energy and creativity.
He supplied two well-taken finishes, the first coming at a crucial time in the game when the Goodison crowd were becoming increasingly frustrated with their side's sloppy play, and might have had the match ball if not for a late goal-line clearance.
Walcott was also unfortunate not to come away with at least one assist to his name with several clever deliveries from out wide going unexploited, but it was his pressing and work rate that was perhaps most impressive.
Winning the ball back on multiple occasions repeatedly endeared him to the home fans and set the tone for his team-mates. He was also their most lively outlet in the final third - with four shots on target.
If he can continue in this vein, it won't be long before Walcott is a highly-valued member of the Everton squad - as shown by the delighted chants of "Theo, Theo, Theo!" which greeted the full-time whistle.
The night belongs to Coleman
The night started with one raucous reception for Coleman and ended with another.
He walked out of the tunnel to be greeted by a Park End flag adorned with the words: "Welcome back Séamus" before his name was purposefully read out last of Everton's line-up in the pre-match build-up, understandably met with a deafening cheer.
And come the full-time whistle - Coleman ignored the driving rain to show his appreciation for the fans' support at the Gwladys Street End, once again met by great acclaim from those who had not yet headed home.
The defender had not featured since March due to a double leg break sustained on international duty with Republic of Ireland against Wales but you wouldn't have thought it here.
The 28-year-old is a popular figure in these parts and proved exactly why with a performance high on enthusiasm and endeavour, constantly driving down the right-hand side and combining with Walcott as if he had never been away.
Two surging runs in particular showcased his superlative fitness, which belied such a lengthy spell on the sidelines, as he beat four or five Leicester defenders across 40 yards before releasing Niasse in one.
In another, a late 80-yard run almost led to a third goal, though he ultimately ran out of legs to play the pass with enough power to reach its target.
Allardyce called the Irishman "an example" to the rest of his Everton team-mates afterwards and to think that he played so well having only managed 58 minutes of action - and that for the under-23s - in the last 10 months before this is truly remarkable.
He has a big part to play for the rest of this season as Everton look to finish as high as possible and potentially do battle for a Europa League spot, with Leicester - three points ahead in the table - one of their likely challengers.
The Mahrez distraction?
That Leicester boss Puel was asked afterwards double the amount of questions about Riyad Mahrez as he was about the game itself spoke volumes about the main story of the night from the Foxes' perspective.
Without the influential Algerian winger after he reportedly decided not to join up with the rest of his team-mates having missed training on Tuesday, Leicester were second best for large periods.
They started the brighter team initially and made greater use of the ball. They almost went ahead when Ndidi was denied by the woodwork, but failed to capitalise when on rop.
Likewise later in the game they didn't have enough quality, though Everton equally defended well, to force an equaliser despite their pressure - hitting the woodwork twice more.
It was Leicester's defending more than anything that let them down on the night with Chilwell guilty of losing possession too easily for both goals.
There is an argument to say that the Mahrez saga - as he failed to force a move through to Manchester City despite a transfer request - took its toll on a Leicester side who had not lost in six before this.
But Puel argued that Leicester still had enough about them to get something from the game.
They probably did, but in general play they failed to find Vardy enough and after Mahrez's replacement Demarai Gray had started well - few Leicester players really performed even despite their late flurry of chances.
Leicester will hope they can reintegrate Mahrez - with eight goals and eight assists in the league this term - as well as they did after he failed to force a summer exit.
That the former PFA Player of the Year is reportedly seeking showdown talks to clarify why he was not allowed to leave will not exactly help matters, but he surely has little option but to get back to business.
Leicester will certainly hope so at least, for they are a different prospect with a fit-and-firing Mahrez available.