Trevor Zegras had two assists for his first NHL multipoint game for the Ducks (11-21-7), who have lost three in a row.
"They were engaged," Anaheim coach Dallas Eakins said. "The work was there. We've just got to, again, bury our chances."
John Gibson returned for the Ducks after missing three games with an upper-body injury and made 28 saves, including a penalty shot against Keller six seconds before Chychrun's game-winner. He is the only NHL goalie this season to face three penalty shots and has made the save on two of them.
"Obviously, we had a good save there," Eakins said. "They ended up winning the draw and scoring. The penalty shot didn't rattle us or throw us off-kilter. I thought [Gibson] did a great job holding his net."
Chychrun tied it 2-2 with a wrist shot from the left circle at 15:09 of the third period.
Sam Carrick made his season debut on an emergency loan from San Diego of the American Hockey League and gave the Ducks a 1-0 lead at 8:31 of the first period. Zegras won the puck behind the net and fed Carrick for a one-timer.
"We called him up knowing exactly what we were going to get from him," Eakins said of Carrick. "It's going to be hard work and nothing less."
Carrick said he felt comfortable on a line with Zegras and Max Jones after he played with each in the AHL.
"Trevor Zegras is a perfect example of a guy that we're trying to get the puck in his hands and let him work his magic," Carrick said. "For his age (20), what he's doing at this level, is pretty special."