The win follows a run of three draws in the league and it moves Jose Mourinho’s men to within 12 points of pacesetters Manchester City who host Watford on Tuesday night at the Etihad Stadium.
Martial, who started in the absence of the injured Romelu Lukaku, broke the deadlock in the 57th minute at Goodison Park. Teed up on the edge of the penalty area by his fellow France international Paul Pogba, Martial curved the ball into the top corner of Jordan Pickford’s goal.
Lingard wrapped up the points nine minutes from time. The England international slalomed past several Everton players before curling home from 20 yards for his fourth goal in his last five appearances.
The three points were a boost for Mourinho amid reports the United manager was losing the support of his players after branding them childish and immature for a series of simple mistakes in the draws against Leicester City, Burnley and Southampton over the Christmas period.
"Our pride, our effort, everything was top class and on top of it we played very well. It wasn't a performance of tired boys, with the worst possible calendar," Mourinho said.
He added: "Jesse Lindgard is more consistent. He is adaptable and he is going in a good direction."
Liverpool are fourth following a last gasp 2-1 win over Burnley. With Philippe Coutinho and Mohamed Salah missing due to thigh and groin injuries respectively and Roberto Firmino left on the bench, Sadio Mane was the lone member of Liverpool's so-called 'Fab Four' to start at Turf Moor.
The Senegalese striker showed his side can thrive without his star teammates. He opened the scoring in the 61st minute. That looked to have sealed the points but three minutes from time, Johann Gudmundsson levelled for Burnley.
However, four minutes into stoppage time, the Merseysiders had the last laugh. Dejan Lovren headed on a free-kick and Estonia defender Ragnar Klavan poked home from close-range.
Liverpool's third successive win extended their unbeaten league run to 13 games and they moved to within one point of third placed Chelsea who play at Arsenal on Wednesday night.
"It was a very difficult game,” said Liverpool boss Jurgen Klopp. “The weather made it so hard. It was windy and rainy. This is one of the toughest places to play all season. Did we deserve to win? I don't care. We got the three points."