For large periods it looked as though the two teams would play out a dull goalless draw despite Jay Rodriguez wasting a good opportunity before half-time and Fraser Forster thwarting Romelu Lukaku on a few occasions.
But second-half substitute Valencia's first goal in a year, and his first for the club, broke the deadlock on 73 minutes as he pounced on a rebound that arrived after Lukaku had been denied by Forster from close-range.
In the final 10 minutes, Baines sent Forster the wrong way from the spot with a club record-equalling 23rd successful penalty after Maya Yoshida's foul on Valencia.
In the final stages, the Belgian fired high for a third to condemn the Saints to their worst losing run in the Premier League for over two years and maintain an unbeaten home run against the south coast club which stretches back to November 1997.
Both sides utilise squad depth
Having spoken of the need to use "other, fresh players" in his pre-match press conference, Claude Puel made at least four changes to his starting eleven for the sixth successive league game.
With influential centre-back Virgil Van Dijk suspended, captain Jose Fonte returned in central defence while Cedric Soares, Jordy Clasie, Jay Rodriguez, James Ward-Prowse and Nathan Redmond, back from a ban, also all came in.
Left-back Ryan Bertrand only made the bench despite being rested for the home defeat to West Bromwich Albion on Saturday, with 21-year-old Sam McQueen preferred for his fourth start of the campaign.
Everton boss Ronald Koeman handed a first Premier League start to 19-year-old forward Dominic Calvert-Lewin as part of four changes, with 18-year-old midfielder Tom Davies given his second league start as they became the first top-flight team this season to name two teenagers in their starting line-up.
Elsewhere, Aaron Lennon and Ross Barkley - fresh from his late equaliser away at Hull City on Friday night - replaced Kevin Mirallas and Enner Valencia as the Dutchman replenished his attack.
Neither side impressive in insipid first-half
The last time these two teams last met, only six weeks ago, Charlie Austin put the Saints head inside just 41 seconds - but the start here was a little more understated, albeit Fraser Forster being required to push away Barkley's half-volley.
Inside just five minutes, the visitors were forced to withdraw Cedric and hand Jack Stephens his league debut after the right-back sustained an apparent head injury in clattering into the advertising hoardings.
And Everton too had to make an early substitution with Calvert-Lewin's debut ended early by an ankle injury which resulted in Mirallas' introduction on 12 minutes.
Those stoppages, and the fatigue of the festive fixture schedule, meant for a largely dull opening half-an-hour. Forster's extended right leg, denying Lukaku's low drive, was the only real chance of note for the home side as they desperately searched for some inspiration.
Southampton had the ball in the back of the net on 26 minutes through Rodriguez, only for the offside flag to rightly be raised.
But it was Rodriguez, making only his fifth start this season after a lengthy injury, who appeared the Saints' likeliest scorer.
The Englishman wasted their best opportunity of the first-half on 39 minutes when he flashed a left-footed effort over the crossbar after breaking beyond Ashley Williams to get into the box.
Meanwhile Everton's sloppy passing, and their continually disappointing final ball limited them to very few chances, Mirallas firing over up the other end before a long-awaited half-time whistle brought a first 45 minutes devoid of quality but full of frustration to an end.
Both teams still flat to begin second-half
Koeman had called Everton's defeat at Southampton on November 26th one of his team's worst performances of the season, but his players threatened to lurch to a new low here as the second-half started in the same fashion.
Seamus Coleman was fortunate that his goalkeeper was quick off his line to clear his under-hit back-pass before the lurking Rodriguez could meet it.
And they were equally as lacklustre at the opposite end of the pitch, Mirallas unable to pick out the run of Lukaku after an encouraging passing move - much to the frustration of the home crowd.
Their complaints steadily grew in decibels as the game progressed and Koeman's response was to introduce Enner Valencia in place of Lennon.
That coincided with an increase in tempo from Everton, albeit one that yielded no immediate reward as Lukaku dragged a shot wide from 20 yards after driving into space.
Southampton's pace on the counter-attack also threatened to catch the Blues out as they went in search of the opener, Robles producing a strong reflex save to keep out Redmond's strike after Rodriguez's flick-on.
Valencia tap-in triggers late flurry of goals
Yet the elusive breakthrough finally came on 73 minutes through substitute Valencia, who scored his first goal for the club.
Forster managed to save Lukaku's downwards header from Coleman's cross at point-blank range, only for Valencia to steal in and score his first club goal anywhere since 23 January 2015.
And just eight minutes later, Everton made sure of the three points after Yoshida hauled Valencia to the ground inside the box.
Baines stepped up to the spot and was as precise as ever from 12-yards, equalling Trevor Steven and David Unsworth's record of 23 successful penalties by sending Forster the wrong way.
Yet, despite having looked so short of ideas before the opening goal, Everton added a third late on to add gloss that their performance - at least for the first 73 minutes - didn't quite deserve.
Baines won the ball deep inside the Southampton half, finding Davies whose first-time pass allowed Lukaku to control and fire a powerful effort into the top corner to seal the victory in style with his 11th goal of the campaign.