Chicago (3-8-6) won for the first time since a 2-1 victory over the Houston Dynamo on April 6.
Gutiérrez intercepted a pass near midfield and raced toward the goal before he rolled a shot from 25 yards out that ricocheted off the post before rolling into the net to give the Fire a 2-1 lead in the 61st minute. The 20-year-old Gutiérrez, in his fifth MLS campaign, has four goals this season, matching his career best.
The Galaxy (7-3-7) had their six-game unbeaten streak ended and lost for the first time since a 2-0 loss to Austin FC on April 27.
Riqui Puig converted from the penalty spot in the seventh minute to give the Galaxy a 1-0 lead, but Cuypers ran on to a failed clearance attempt and poked a sliding shot into the net from point-blank range to make it 1-1 in the 38th.
Chris Brady had three saves for Chicago. Novak Micovic had two saves for the Galaxy in his second career start.