The France midfielder's deflected volley in the 71st minute was enough for a hard-earned victory in a match full of VAR controversy.
United move three points clear of Liverpool ahead of their showdown at Anfield on Sunday.
Not for the first time in the evening, VAR took centre stage in the 32nd minute.
After referee Kevin Friend initially showed the Burnley man a yellow card, the incident was reviewed by VAR.
Following several replays, the referee instead awarded Burnley a free-kick, showing United defender Shaw a yellow card and so the possible red card against Brady was ruled out, with his caution rescinded.
Soon after the restart, Cavani spurned a glorious opportunity to break the deadlock when he scuffed his shot straight at Pope from close range after being picked out by Bruno Fernandes.
United eventually made the breakthrough in the 71st minute.
Marcus Rashford got clear down the right and clipped the ball back across to the edge of the penalty area, where Pogba sent a volley goalwards which deflected off Burnley defender Matthew Lowton and past Pope.
VAR was again in action during the closing stages for a review of a potential handball by Maguire in a crowded six-yard box, but no penalty was given.
Earlier, bottom club Sheffield United finally ended their long wait for a Premier League victory after they beat 10-man Newcastle 1-0 at Bramall Lane.
The Blades, who had won in the FA Cup third round on Saturday, picked up all three points for the first time this season following Billy Sharp's penalty in the 73rd minute.
Newcastle lacked imagination and their task was made more difficult following the dismissal of winger Ryan Fraser just before half-time for a second caution.
Sheffield United – whose last Premier League win came on July 11 – are up to five points, three behind West Brom, but having played more games than their relegation rivals.
Everton are fourth after they beat Wolves 2-1 at Molineux.
Wolves, though, equalised in the 14th minute through Ruben Neves' volley.
With 20 minutes left, Neves went close when he fired a spectacular long-range effort which dipped just over.
It was Everton, though, who regained the lead in the 77th minute when Michael Keane headed in at the back post.