This means that Man United is in the league lead after New Year's for the first time since 2013 and Sir Alex Ferguson's glory days.
On Sunday, Liverpool await in a league final at Anfield.
It was the star players who were behind the goal.
Bruno Fernandes unleashed a swerving effort from the right that the visitors' keeper did well to tip wide.
In the final minutes, Burnley had a great chance to equalize but James Tarkowski failed to push down the bouncing ball.