Ole Gunnar Solskjaer has now won 10 of his 11 games in caretaker charge of United, while Fulham remain in 19th position and seven points adrift of safety.
Fulham should have been ahead within the first 60 seconds of the game, but after some good work from Andre Schurrle down the right flank Luciano Vietto somehow failed to hit the target at the back post after meeting the German's cross.
Schurrle then saw a shot deflected over the crossbar from near the penalty spot, but those missed opportunities allowed United to settle and the visitors went ahead just before the quarter-of-an-hour mark.
Pogba ran on to a pass from Martial inside the penalty area and appeared set to pull the ball back for a teammate, but the Frenchman instead drilled the ball inside the near post with Sergio Rico having committed himself to trying to claim a cross.
United only grew in confidence as the first half progressed, but their second was down to the individual brilliance of Martial who ran from the halfway line and saw off the attention of two Fulham players before finding the far corner from 15 yards out.
Fulham managed to regain their edge in the middle of the park in order to prevent United from further extending their lead, but Phil Jones almost added a magnificent third for the visitors with his left-footed strike from 25 yards being tipped over by Rico.
Calum Chambers saw a volley from distance saved by David de Gea just before the half-time whistle, but Fulham went into the break needing to overcome a two-goal deficit for the second home game in succession.
Claudio Ranieri opted to change to a back three shortly after the restart, but the withdrawal of Schurrle was widely booed by the home supporters and it set the tone for the remainder of the match.
The Cottagers barely posed a threat at one end, but United were going through the motions at the other and Ander Herrera should have added a third with a 15-yard volley only for the ball to be directed straight at Rico.
The game was effectively over as a contest, but Fulham came close to pulling a goal back when Aleksandar Mitrovic saw a header kept out by De Gea down to his left.
United responded with Alexis Sanchez capitalising on a mistake in the Fulham backline before being denied by Rico, while fellow substitute Scott McTominay sent a glancing header marginally over the crossbar.
Ryan Babel then failed to score from little more than a yard out at the back post, and that missed opportunity summed up Fulham's day and season with Ranieri struggling to find a solution to his side's issues at both ends of the pitch.
The closing stages of the match were completed with minimal action inside either penalty area, and United will now focus their attention on trying to earn a result against Paris Saint-Germain in the first leg of their Champions League last-16 tie on Tuesday night.
FULHAM (4-4-2): Rico; Odoi, Le Marchand, Ream, Bryan (R.Sessegnon 81'); Schurrle (Christie 53'), Chambers, Seri, Vietto; Babel (Cairney 77'), Mitrovic
MANCHESTER UNITED (4-2-3-1): De Dea; Dalot, Smalling, Jones, Shaw; Herrera (Bailly 85'), Matic; Mata, Pogba (McTominay 74'), Martial (Sanchez 70'); Lukaku