The win sees Manchester United extend their unbeaten run to 11 games and move atop the Premier League for the first time this late in a campaign since December 2012 when Sir Alex Ferguson was manager.
After the match, Pogba said: "We know that the game today was very hard, very difficult and obviously we wanted to win. We know it's not easy to play here. We got the three points and we're happy for that, but there's still a long way and now we focus again until the end of the season."
A double VAR review near the half-hour mark saw referee Kevin Friend show United defender Luke Shaw yellow for a foul on Johann Gudmundsson, while waving off a caution initially given to Burnley's Robbie Brady at the other end for a foul on Edinson Cavani.
The hosts stood tall for the rest of the first half, with Nick Pope doing well to parry away an Anthony Martial shot from distance shortly before the break as both teams went to the dressing room goalless.
Burnley stood firm again throughout the first 25 minutes of the second half before Pogba scored a magnificent goal, volleying past Pope with his first touch from a Marcus Rashford assist.
The goal was Pogba's first in all competitions in his last eight games for Manchester United.
United saw out a nervy end to the match as Burnley pushed for the equaliser, but they held on for all three points and have not lost away from home in the league since Jan. 19, 2020, against Liverpool, who they'll face in a table-topping clash on Sunday.
Pogba added: "Always happy when I win -- today we won and I'm happy we play well. Still a long way to go and we have big games to go, so we have to focus on that."