The victory, in the re-arranged fixture, put United on 36 points from 17 games with rivals Liverpool on 33 points ahead of the clash between the two sides at Anfield on Sunday.
The breakthrough came in the 71st minute when Marcus Rashford floated in a ball from the right which Pogba met with a volley that deflected off Burnley defender Matt Lowton and beat the wrong-footed Nick Pope.
It is the first time United have been top of the league after 17 games, since December 2012.
United skipper Harry Maguire had an effort ruled out in the first half after referee Kevin Friend ruled he had pushed Erik Pieters as he rose to head in a Luke Shaw cross at the back post.
Burnley applied some late pressure and felt they should have had a penalty when the ball appeared to hit Maguire's arm and James Tarkowski had a great chance to grab a point for the Clarets but failed to make good contact from a goalscoring position.