Having won just one of their previous five Premier League games, the pressure was on United going into Sunday's clash.
Manager Jose Mourinho opted to drop the out-of-form Romelu Lukaku for the visit of his former side, but his stand-ins did enough to have all but secured his side the victory before his second-half introduction.
It was a nip-and-tuck first 45, with both sides having chances in the opening exchanges.
Andre Gomes and Juan Mata both planted headers straight into the hands of David de Gea and Jordan Pickford respectively, before the game turned in the home side's favour on 27 minutes.
Idrissa Gueye was adjudged to have brought down Martial illegally inside the box, despite replays appearing to show that he got a foot on the ball.
Jon Moss pointed to the spot and whilst a stunning save from Pickford away to his right denied Pogba's initial spot-kick, the Frenchman was able to side-foot home the rebound to put United in the driving seat.
Marcus Rashford, playing through the middle after Lukaku was relegated to the bench, thought he had doubled the lead just minutes later, but he was correctly flagged offside as he converted from close range after Pickford had denied Pogba.
Gylfi Sigurdsson once again headed into the hands of de Gea as the visitors tried to respond, but United went in at the break with a one-goal lead.
It didn't stay that way for long though as Martial doubled the deficit four minutes after the restart with the goal of the game.
Pogba teed up his fellow countryman on the edge of the area and he duly swept a wonderful curling effort beyond the dive of Pickford and into the far corner.
Everton should have got themselves back into the game within two minutes, but Bernard could only find the side-netting after rounding de Gea.
Pickford's impressive performance continued as he made sharp saves to deny Rashford and Pogba, before the latter played a part in handing Everton a lifeline.
The scorer of the opening goal over-complicated things in the middle of the park and surrendered possession, with the Everton breaking ending as Chris Smalling scythed down Richarlison in the box.
Sigurdsson confidently dispatched the resulting penalty, but it proved only to be a consolation as United successfully shut up shop in the final 13 minutes.
Indeed, it was Mourinho's side that had the better chances late on, Lukaku heading wide at the far post after being found unmarked from Pogba's right-wing cross and Martial being denied by Pickford after racing clear having been granted the freedom of Old Trafford.
The win sees United leapfrog Everton into eighth, five points off the champions League places, with the 'Toffees' now sitting in ninth.
October 28, 2018