Arsene Wenger's men looked set to drop two crucial points after Andre Gray converted from the spot, before the Chilean saved the day.
The skipper made history by scoring his 250th goal for the Red Devils in the 94th minute of stoppage time as the visitors extended their unbeaten run.
Los Blancos got back to winning ways in La Liga on Saturday afternoon after losing their previous two games in all competitions.
The Poland international was on target in the first half to cancel out Haberer's opener and struck again in injury time to decide the encounter.
Jonny and Iago Aspas made the difference for the visitors as they hit either side of Marcelo in a frantic six-minute spell that decided the encounter.
The Swede came to the rescue of team-mate Paul Pogba, who had been cast as the villain of the piece with an awful handball to give away a penalty.
The Uruguayan scored twice while Lionel Messi, Arda Turan and Aleix Vidal were also on the scoresheet as the Blaugrana eased to a comfortable victory.
The usual star man was on show for Mauricio Pochettino's men as they strolled to three points to make it six Premier League wins in succession.