Kenyan midfielder Wanyama headed past Victor Valdes in the 21st minute with Celtic's only effort of the first half while Lionel Messi and Alexis Sanchez were both denied by the woodwork.
Celtic goalkeeper Fraser Forster repulsed Messi in the second period before Watt latched on to a long clearance from Forster to score in the 83rd minute.
Messi netted in the first minute of extra time but it was too late for Barcelona who slipped to just their second defeat under Tito Vilanova.
Barcelona controlled the ball from the start and were close to an early breakthrough when Xavi's shot was deflected by Alexis Sanchez only for Forster to make the save.
Against the run of play, the hosts took the lead with their only effort on goal in the opening period. Charlie Mulgrew's corner to the far post was met by Wanyama, who climbed higher than Jordi Alba to head in.
Messi came close to an equaliser when he cut infield and swapped passes with Andres Iniesta before his shot glanced against the bar.
Daniel Alves also sent a cross to the far post and Sanchez's header beat Forster but came back off the inside of the post.
The hosts managed a second chance just before the hour when Mikael Lustig's header was easily gathered by Victor Valdes.
But Forster pulled off a fine double save from Sanchez after the Chilean winger had been played through by Messi.
Argentina forward Messi saw a thumping effort pushed away by Forster and Iniesta's 25-metre effort flew well wide of the goal.
With five minutes remaining, Xavi missed Forster's long clearance and 18-year-old Watt broke clear to score with a low shot. The visitors finally scored when Pedro's effort was pushed away by Forster and Messi touched in during added time.
