Kesler lifted a rebound high in the net at 5:52 of the first period to give Anaheim a 1-0 lead.
Kesler, 34, sat out the preseason and first three games to gain strength and conditioning related to his hip surgery in June 2017. He played the final 44 regular-season games and all four Stanley Cup Playoff games in the sweep by the San Jose Sharks in the Western Conference First Round last season, but wasn't 100 percent.
"It was nice to hear the cheers, and the welcome back that I got was pretty special since everything I went through for past year and a half," Kesler said. "It hasn't been fun, and I had fun out there tonight."
Strome scored on the power play to tie it 1-1 at 17:33.
Arizona went on the power play after Anaheim defenseman Andrej Sustr was called for hooking Brendan Perlini.
Strome dove against the wall and nudged the puck back to Clayton Keller at the blue line to keep possession in the Anaheim zone, and Keller passed back to Strome as he got to his feet. Strome had room to skate into the left circle and take a wrist shot that beat Gibson between his pads for the first goal of the season for the Coyotes.
"It was a lot of relief," Strome said. "Everyone was kind of just like oh, thank God. We definitely didn't want to go into the end of a period, another period, without scoring a goal."