United stars Bruno Fernandes and Marcus Rashford sealed the points with first half penalties at Old Trafford.
Erik ten Hag's team are now eight points adrift of fourth placed Aston Villa and three behind fifth placed Tottenham in the race to qualify for next season's Champions League.
Everton sit four points above the relegation zone as their winless league run extended to 11 games.
Back-to-back defeats to Fulham and Manchester City had put a huge dent in United's top four ambitions.
With Villa and Tottenham facing off on Sunday, they made the most of Everton's mistakes to close on their rivals.
Alejandro Garnacho was the difference between the sides as his quick feet twice provoked spot-kicks.
James Tarkowski was the first Everton defender lured into a mistimed challenge on Garnacho after 10 minutes and Fernandes rolled home his eighth goal of the season.
Garnacho danced through a series of challenges in the 36th minute before he was chopped down by Ben Godfrey.
Fernandes passed over penalty duties to Rashford and Jordan Pickford was unable to deny the forward, who fired home his fifth goal in nine league games.
Saluting Garnacho's impact, Ten Hag said: "I love to work with Garnacho and many other players in the squad but he is a player who likes a challenge.
"Brave, confident and our job is to push him to high level but he is doing this. He has high potential."
With leaders Liverpool facing second placed Manchester City in a blockbuster clash at Anfield on Sunday, Arsenal have the opportunity to move into first place for 24 hours at least.
Third placed Arsenal would move one point clear of Liverpool if they beat Brentford at the Emirates Stadium in Saturday's late game.
Mikel Arteta's side have won their last seven league matches, scoring 31 goals in the process, to boost their bid for a first title since 2004.
Luton earn priceless point
Third bottom Luton rescued a priceless 1-1 draw at Crystal Palace thanks to Cauley Woodrow's last-gasp leveller.
Oliver Glasner's side struck in the 11th minute when Luton's Alfie Doughty played a poor back-pass and Daniel Munoz teed up Jean-Philippe Mateta to flick home.
But Woodrow levelled from Andros Townsend's cross six minutes into stoppage-time to leave Luton three points from safety.
Bottom of the table Sheffield United blew a two-goal lead as Bournemouth hit back to rescue a 2-2 draw at the Vitality Stadium.
Bournemouth's Dominic Solanke missed a 14th minute penalty awarded for a foul by Tom Davies, the striker slipping in his run-up and ballooning the spot-kick over the bar.
Gustavo Hamer put United ahead in the 28th minute with a close-range finish after Jayden Bogle's shot was parried.
Jack Robinson bagged the Blades' second in the 64th minute, netting from close-range after Bournemouth keeper Neto punched the ball against Solanke.
But Dango Ouattara got one back with a header from a 74th minute corner and Bournemouth equalised in stoppage-time when Enes Unal stabbed home to claim his first goal for the club.
Wolves improved their prospects of qualifying for Europe next season with a 2-1 win against Fulham at Molineux.
Rayan Ait Nouri lashed into the roof of the net from Toti's cut-back into the area in the 52nd minute.
Nelson Semedo's long-range shot took a hefty deflection off Tom Cairney for a 67th minute own goal and Alex Iwobi's stoppage-time effort was no consolation for Fulham.
