Both sides came into this game on the back of disappointing results as United conceded a stoppage time goal to draw 2-2 at home against Southampton, while Palace lost 2-0 at relegation-threatened Aston Villa.
Injuries to first-choice left-backs Luke Shaw and Brandon Williams meant United boss Ole Gunnar Solskjaer handed a first start to Timothy Fosu-Mensah since 2017 in one of two changes following three successive matches with the same starting XI.
"We need to smile and go into this, we have earned this right, no-one thought in January that we would be in with a shout for top four," said Solskjaer, whose team are 19 unbeaten on all competitions.
"We have done fantastic since the restart. These are the games that matter. That is the thing at Manchester United, you have to step up and show your personality," he said.
The opening exchanges were scrappy with neither side really taking control, though United captain Harry Maguire wasted two chances when he got his head to first-half corners.
With the first half coming to a close, Palace winger Wilfried Zaha felt aggrieved at not having been awarded a penalty following a challenge by Victor Lindelof and, while VAR checked the incident, United went down the other and to open the scoring.
United talisman Bruno Fernandes played in Rashford who feinted to shoot before turning back on himself to tuck the ball home for his 22nd of the season.
Palace thought they had an equaliser 10 minutes into the second half which Jordan Ayew slid home at the back post following Zaha's lay off, only for the VAR to rule it was marginally offside.
As Palace pushed for an equaliser, United started to find space as Fernandes hit the post from close range after being teed up by Rashford.
United put the game to bed on 78 minutes as Martial showed great strength to finish off following intricate play from Fernandes and Rashford. The goal meant Martial also reached 22 for the campaign.
"It was a tight game and we didn't get the rhythm in our passing but two fantastic goals won it for us," said Solskjaer.
Palace defender Patrick van Aanholt was seriously injured in the trying to defend the goal and had to receive oxygen before being stretchered off.
Despite 11 minutes being added on due to the injury, United held on and moved back level on points with fourth-place Leicester City, who had beaten Sheffield United 2-0 earlier in the day. Palace stay 14th.