Returning from suspension, the former Manchester United star scored his first goals since mid-April, but his side could not help his team avoid its seventh loss in 13 outings this season.
Ryan Hollingshead completed a nice team move to give Dallas the lead, which was doubled before halftime when Cristian Colman headed in.
A goal from Carlos Gruezo put the visiting side in complete control 66 minutes in.
But Ibrahimovic – earlier denied by the offside flag – drilled in his fourth MLS goal three minutes later to give his side a bit of hope.
Colman saw red with 10 minutes remaining, and the Swede was on hand to turn in another finish late on.
But the Galaxy were unable to find an equalizer and failed to continue their winning run in league play.
For Dallas, the victory takes the club up to second in the conference, just three points back of Sporting Kansas City with a game in hand. It also marked the club's 100th win away from home in MLS play.
Josef Martinez opened the scoring for Atlanta with his ninth of the season, but the side was pegged back late.
Teal Bunbury converted an 88th-minute penalty for the Revolution, who are sixth in the east.