Sporting Kansas City (4-5-7) won for the second time in 12 matches (2-4-6) and moved to 1-3-4 on the road.
Columbus (5-10-2) has four losses and a tie in its past five games and is 1-9-1 in 11.
Gutierrez scored in the 48th minute but the work was done by Gerso, who raced to the end line to the right of the net to keep the ball in play. He nutmegged Luis Argudo then serviced the ball for the Gutierrez header on SKC's first shot on goal.
Melia had 23 shutouts the previous two seasons, including 13 in 2018, but his lone one this season was March 10 against Philadelphia. He has 41 in his MLS career.
The Crew was missing Gyasi Zardes, who scored twice for the U.S. national team against Trinidad and Tobago on Saturday, and was blanked for fifth time.
Columbus went down a man in the 74th when Harrison Afful denied Gerso a scoring opportunity.
Sporting Kansas City, which had as many as 10 players injured at one point this season, lost Johnny Russell to a hamstring issue during training on Saturday. His seven goals tie him for the team lead with Gutierrez and Krisztian Nemeth.
The Crew lost midfielder David Accam to a leg injury in the 16th minute.
Melia made a pair of saves in the first half, starting with an easy stop of a 15-yarder off the foot of Hector Jimenez in the 22nd. Afful tested Melia in the 36th minute from the top of the box to no avail.
On Saturday, the Crew hosts Orlando City while SKC plays at Real Salt Lake.