It marked the club's first ever Premier League triumph at Old Trafford.
Chris Wood, who also struck, added: "We had to grind it out. Luckily we got the two goals which gave us a bit of comfort."
And Burnley skipper Ben Mee said: "It's nice to see the game out and keep a clean sheet. It's a funny game. We lost four league games in a row and now we've beaten Leicester and Manchester United in the space of a few days."