The visitors edged a tense first 45 minutes at the Nou Camp, but it was the hosts that took the lead after the second half through former Chelsea star Marcos Alonso after 50 minutes.
But Manchester United were quickly back level after Marcus Rashford scored his fourth goal in four games in the 52nd minute to make a quick riposte.
And the Red Devils then took the lead just seven minutes later after Jules Kounde turned the ball into his own net on 59 minutes to turn the game around on the night.