United now lead defending champions Liverpool by three points ahead of a top-of-the-table clash between the two most successful sides in English football history at Anfield on Sunday.
Despite continued speculation over his future, Pogba has played an increasingly important role in United's first serious title challenge since legendary former manager Alex Ferguson retired in 2013.
Ole Gunnar Solskjaer's men have now taken 29 from the last possible 33 points to recover from just two wins in their opening six games of the season.
The signing of Bruno Fernandes shortly after Burnley won 2-0 at Old Trafford nearly a year ago in the last meeting between the sides has transformed United's fortunes.
And if Pogba can finally produce consistent displays worthy of the then world record fee United paid for the Frenchman in 2016, there is reason for Solskjaer to believe he can end the club's eight-year wait for a league title.
The manner of victory against a Burnley side that had not lost at home since October also suggests United have the ability of champions to grind out a result when they need to.
After a slow start by the visitors, the first-half was dominated by controversy over a series of VAR reviews.
A five-minute delay to review Luke Shaw's challenge on Johann Berg Gudmundsson and Robbie Brady's foul on Edinson Cavani as the Uruguayan bore down on goal resulted in both sides escaping without having a man sent-off.
Referee Kevin Friend had initially waived play on from Shaw's lunge, but on review the left-back was shown a yellow card and Brady's yellow rescinded as the game should have been stopped earlier in the move.
United then had a goal harshly ruled out when Harry Maguire powered home a header at the back post, but was punished for a slight nudge on Erik Pieters.
The visitors upped the pressure in the second-half as Cavani, back from a controversial three-game ban, missed two big chances he would expect to take.
Pogba was controlling the game from midfield and it was fitting he was the match winner as his volley from Marcus Rashford's cross deflected off Matt Lowton and through the legs of Pope.
Burnley had chances to equalise late on as Eric Bailly diverted Matej Vydra's shot just wide before Maguire escaped claims for a penalty for handball.
But United held on for a big psychological boost ahead of the trip to Anfield.
Everton win at Wolves
Everton moved back into the top four and within four points of United as they overcame an injury crisis to beat Wolves 2-1 at Molineux.
Carlo Ancelotti was forced to start without a recognised striker without Dominic Calvert-Lewin and Richarlison only fit enough for a place on the bench.
The Toffees took an early lead when Alex Iwobi converted Lucas Digne's cross but were pegged back when Ruben Neves levelled for Wolves.
Michael Keane's towering header won the game 13 minutes from time to bolster the Blues' European ambitions.
At the other end of the table, Sheffield United ended a 20-game winless league run as Billy Sharp's penalty earned a 1-0 victory over 10-man Newcastle.
The Blades still face an uphill task to haul themselves off the bottom of the table as they sit nine points adrift of safety.
But Chris Wilder's men ensured they will not set an unwanted record by going a full top-flight season without a win thanks to another night to forget for Newcastle.
"It looked like we got our identity back," said Wilder.
"I felt we had an identity and it was us with the handbrake off, going for it and playing well in and out of possession."
The Magpies could yet find themselves dragged into a relegation battle as they remain eight points above the drop zone in 15th, but have played two games more than Fulham in 18th.