United boss Jose Mourinho dropped Romelu Lukaku to the bench amid a difficult run of form for the Belgium international, and the Red Devils responded by collecting all three points at Old Trafford despite a late scare.
The result has seen the 20-time English champions move above Everton into eighth position in the Premier League table ahead of their trip to in-form Bournemouth next weekend.
Pogba had the first shot of the match inside the first three minutes, but the Frenchman's effort from outside the Everton box went out for a throw-in.
United had plenty of the ball in the first 10 minutes, but Everton were threatening on the break, with the pace of Richarlison, Bernard and Theo Walcott causing the home side's defence plenty of problems.
Everton's first chance of the match arrived in the 14th minute when on-loan Barcelona midfielder Andre Gomes found all sorts of space inside the United box, but his header was straight into the arms of David de Gea.
Martial then tested Pickford with a powerful effort in the 23rd minute as opportunities continued to arrive.
Martial was United's most dangerous attacker, and the Frenchman won his team a penalty in the 26th minute when he went down under a challenge from Idrissa Gueye.
Pogba took an age in his run-up, which brought boos from the crowd, before seeing his penalty saved by Pickford, but the rebound fell kindly for the Frenchman, who stroked the ball into the opposite corner to send United ahead in the 27th minute.
Marcus Rashford looked to have doubled United's lead three minutes later when he tapped home at the far post, but the England international was rightly flagged offside.
The Red Devils were having all of the ball as the first period approached its latter stages, and Fred saw a long-range strike just miss the post as Everton continued to come under fire.
The visitors were still having joy in wide areas, however, and De Gea had to keep out a Gylfi Sigurdsson header in the 37th minute, before the United goalkeeper saved from Walcott late in the half as the hosts took a 1-0 lead at the half-time interval.
De Gea made a smart save to deny Richarlison early in the second period as Everton started on the front foot, but United had their second of the match in the 49th minute.
Martial had terrorised the away side in the first period, and it was the excellent Frenchman that doubled United's lead when he curled a sensational effort into the bottom corner following more good work from Pogba.
Everton should have halved the deficit in the 51st minute when Bernard broke into the United box before rounding De Gea, but the Brazilian could only find the side-netting.
Pickford then had to keep out strikes from Rashford and Pogba down the other end as a dominant United looked for a third in front of their vocal supporters.
Lukaku came off the United bench in the 65th minute as a replacement for Rashford during a period where Everton were threatening, and De Gea had to keep out a header from Gomes in the 66th minute.
Seamus Coleman had an even better chance for the Toffees 20 minutes from time after being released by Richarlison inside the United box, but the right-back fired over the crossbar.
Everton were back in the match in the 77th minute, however, when Sigurdsson fired a penalty past De Gea after Chris Smalling had brought Richarlison to the deck with a late challenge.
There were once again eyes on Pogba, with the midfielder needlessly giving the ball away in the middle of the park moments before Richarlison won Everton their spot kick.
Everton were on the front foot in the period that followed, with Richarlison passing up a good chance in the 83rd minute as the tension inside Old Trafford continued to rise.
There were six minutes of additional time at Old Trafford, and United should have scored a third in the fourth of those when Martial broke into the Everton box, but Pickford made a smart save as it finished 2-1 at Old Trafford.
MANCHESTER UNITED (4-3-3): De Gea; Young, Smalling, Lindelof, Shaw; Fred (Lingard 95'), Matic, Pogba; Mata (Herrera 85'), Rashford (Lukaku 65'), Martial
EVERTON (4-2-3-1): Pickford; Coleman, Zouma, Keane, Digne; Gueye (Tosun 78'), Gomes; Bernard (Lookman 62'), Sigurdsson, Walcott (Calvert-Lewin 78'); Richarlison