After a dour first half at Goodison Park, Martial broke the deadlock before the hour mark with a lovely curling effort.
And a similarly-excellent goal from Lingard in the final 10 minutes denied the Blues any hope of a comeback, condemning them to back-to-back defeats for the first time since the end of October.
United's win - their first in five league games after three successive draws - lifts them back above Chelsea, who travel to Arsenal on Wednesday night. Everton remain ninth but have slipped three points behind eighth-placed Leicester City.
Few talking points in dull first 45
The first half was devoid of any creativity or quality with no real opportunities of note for either side.
Paul Pogba was often the architect for United's sparing bright moments, a threatening low cross after a one-two at the corner of the box early on followed by a 25-yard shot narrowly deflected past the post and a clever pass to release Lingard into space before drilling wide of the mark both later in the half.
Marcos Rojo's fine last-ditch tackle denied Oumar Niasse a chance at testing David De Gea before the Argentine forced the first, and only, save of the first half with a tame header that never troubled Jordan Pickford.
But the 'keeper was fortunate that no United player was in a better position to react to a low Martial cross which he could only palm into the six-yard box a minute later.
Wayne Rooney, occupying a deeper role when Everton were in possession, lost the ball to allow Ander Herrera an attempt from distance, though his shot flew well over as was the theme of the first 45 minutes.
Yannick Bolasie's bright run down the left, skipping beyond Nemanja Matić and then Rojo, displayed Everton's potential to get at the visitors but Nikola Vlašić could not meet his cross and that was Everton's only real dangerous move of the opening period.
Martial stunner puts Red Devils ahead
But United were by far the better side after the break with Juan Mata's angled 20-yard strike beating Pickford but going wide courtesy of the outside of the post.
Pogba once again came closest to forcing an opening for the visitors, though Martial could not apply a decisive touch to his square ball as United pressed for the game's opening goal.
But the Frenchman could apply the kind of class that the game had been crying out for on 56 minutes, when he received Pogba's pass on the edge of the box, controlled and quickly curled an effort into the top corner.
That invited a spell of sustained pressure from José Mourinho's side, with Pogba forcing a strong reflex save from Pickford after breaking into space and then just missing out on heading in Martial's cross.
Lingard provides the killer blow
Although Bolasie and Rooney were replaced for Everton, the Blues improved and went close to equalising through Niasse after Pogba had been dispossessed.
The Senegal striker found Mason Holgate out wide and darted back to a central position to receive the follow-up cross, but could only guide his header wide with Everton's best chance of the game.
The hosts' resurgence was brief and after Pickford tipped a low Lingard drive wide, the in-form midfielder scored his seventh goal in his last nine league outings with a goal of similar quality to Martial's.
The midfielder cut inside and used Ashley Williams to bend a delightful strike which curled past Pickford to ensure a much-needed three points for the away side.
United have now won more top-flight matches against Everton (35) than any other side has won against a single opponent and have not lost to the Merseyside outfit in seven successive cup and league meetings.
Blues lack the spark to open up Pogba-inspired United
Allardyce has criticised Everton's creativity of late, insisting in the aftermath of their late defeat at Bournemouth on Saturday that he expected better with their available quality.
And no doubt he will have been disappointed with his side's dreadful use of possession again here.
On numerous occasions Everton squandered momentum with poorly-picked, or poorly-timed, passes in midfield and in the final third.
And their sloppy passing left them vulnerable on the counter-attack, just as it did at Bournemouth. It was a poor attempt from Rooney - whose hard-working yet sloppy and substandard performance epitomised Everton's display as a whole - to look inside that United countered to score their goal.
Herrera, Mata and then Pogba carried the ball forward before the sublime vision of the latter, who was excellent throughout and named Sky Sports' Man of the Match, helped set up his compatriot Martial.
The threat of Pogba, who had demonstrated his play-making abilities with a number of raking cross-field passes throughout the evening, doubled when he took up an advanced left-sided role, and he also set up Lingard's second.
A lot of criticism towards Everton this season has been aimed at their failure to replace Romelu Lukaku, denied a Goodison return here due to a head injury, but even if they complete a deal for Besiktas' Cenk Tosun - they will need to provide him with more service than they offered Niasse and his fellow forwards.
Niasse was left to chase down long balls for much of the game and while his work-rate never dipped, his pressing could not panic United into mistakes at the back. A bright first-half run from Bolasie aside, they did very little to excite a strangely-quiet Goodison crowd.
They actually improved for a 10-minute spell after the substitutions of Rooney and Bolasie, even though the pair were Everton's two most likely creators in the first half, but still struggled to forge enough real opportunities besides Niasse's header.
And their brief spell amounted to nothing with Lingard's goal putting the game past the home side.
The smattering of boos which greeted the final whistle - after Everton failed to produce a single shot on target - said everything about a fan base quickly growing tired of dour attacking displays.
In-form Lingard among the goals again
Mourinho recently claimed that they could not afford to rest Lukaku, but it is Jesse Lingard who is proving the most crucial player to this United team of late, perhaps even ahead of creative lynchpin Pogba.
Lingard's 10th goal of the season here means only Lukaku, on 15, remains ahead of him within the United squad and he is only two away from doubling his best-ever tally in a season - managing six in 40 in 2015-16.
This term already he has 10 in 26, with seven goals and four assists in the Premier League - more than the likes of Alexis Sánchez (seven goals, three assists) and Eden Hazard (five goals, two assists).
More impressively, operating from the left side his tally of six goals since the start of December is only bettered by Tottenham Hotspur's Harry Kane, who scored two hat-tricks and eight altogether in the final month of 2017.
Lingard declared to the Sky cameras afterwards that his belief and drive is behind his recent run of form, having now scored in four of United's five last away games - and it shows.
But Mourinho also deserves credit for Lingard's development, having developed from a bit-part player with occasional use to an important attacker with a reliable output.
He has also developed a knack for goals of the highest quality, as he once again evidenced here, while his work-rate, pressing, movement and game intelligence are all other obvious qualities.
His manager said: "They jump or they don't jump. It's one thing to be a talent, it's another thing to be a quality player. Jesse made that jump. He's been more consistent, more adaptable. He understands better the different spaces and different positions."
At this point Lingard has made less of a jump and more of a leap, and United will hope that his exceptional form continues.