United dominated a contest in which Burnley failed to register a single shot on target until stoppage time.
But they were struggling to make a breakthrough until Marcus Rashford picked Pogba out with an excellent cross to the edge of the area.
The Frenchman’s connection was perfect, although it took a deflection off Matthew Lowton to ensure the ball went past Nick Pope and into the Burnley net.
Although Burnley had three decent chances in a frantic ending, United secured the win to head the table after 17 rounds of matches.
It is the first time that has happened this many games into the season since December 2012, in Sir Alex Ferguson’s final, title-winning campaign.
New territory for post-Ferguson United
In the seven seasons since Ferguson retired, United have never been higher than second at this stage of the season.
But they were already well adrift of Manchester City during that 2017-18 campaign under Jose Mourinho.
Surprisingly, in points terms, the nearest they have been to the top at this stage was under David Moyes, when they were eighth but only eight points behind the leaders after 17 games.
They made hard work of their ninth win in 11 Premier League games. On his return from a three-match ban, Edinson Cavani wasted two excellent chances and Bruno Fernandes fired a first-half effort straight at Nick Pope.
Anthony Martial was off target with two first-half chances before failing to convert in the final seconds after he had been sent clean through.
None of this matters now though and, while manager Ole Gunnar Solskjaer was right to say nobody remembers what the table looks like in January, Pogba’s clenched fist as Burnley’s last chance went begging indicated the importance of these three points.