Danton Heinen scored in the first period for the Ducks (3-2-2), who have a point in each of Gibson's past four starts.
He is 2-0-2 with two shutouts and four goals allowed since Jan.18, and is 8-2-3 with two shutouts in 14 games against Arizona. It was his 21st NHL shutout, and his second this season by a 1-0 score.
"I mean he's unbelievable. It shows why we have so much confidence in him back there," Heinen said of Gibson, who shut out the Minnesota Wild 1-0 on Jan. 18. "He just keeps you in games. He's a special goalie."
Darcy Kuemper made 23 saves for the Coyotes (2-4-1), who lost 1-0 at home for the second game in a row. They lost to the Vegas Golden Knights on Sunday, when Marc-Andre Fleury made 16 saves and William Karlsson scored with 43 seconds remaining.
"[Gibson] is a great goalie but we've got to get in front of the net, try to get some shots through, some hard shots," Arizona coach Rick Tocchet said. "We had the puck a lot tonight, but you've got to find a way to support it. Bottom line, we need some guys to score some goals."
The teams finish the two-game series at Gila River Arena on Thursday (9 p.m. ET; FS-A PLUS, KCOP-13, NHL.TV).
Heinen stole the puck from Coyotes forward Drake Caggiula in the right face-off circle, looked briefly for an open teammate, then put a wrist shot under the crossbar at 10:59 of the first period for his first goal of the season.
"I kind of tried to force it from there earlier, but it came back to me, so I thought just put it on net and luckily it went in," Heinen said. "It's tough to score in this league. You've got to keep on shooting, keep on going to the net."