The Coyotes scored first for the third time in this series when Chychrun made it 1-0 at 2:50 of the first period on Arizona's first shot. Thomas tied it 1-1 75 seconds later on a backhand from below the goal line.
Chychrun scored an empty-net goal with 26 seconds remaining for the 3-1 final.
Arizona limited St. Louis to 14 shots over the final two periods.
"The guys did a good job in front of me," Kuemper said. "We were good at all ends of the rink. Power play was working, [penalty kill] was working, everything that we wanted. It was a huge win."
Thomas left the game with 10:24 remaining in the first period after falling on his left wrist after being tripped by Coyotes forward Nick Schmaltz. Blues coach Craig Berube did not have an update on his status.
"He had a great shift there with [Ivan] Barbashev and [Oskar] Sundqvist and yeah, it's unfortunate," Berube said. "I feel bad for Robert, really bad. He's had a tough start (to the season), but I was convinced he was going to work his way out of it, and now it's an injury, so it's unfortunate."
NOTES: Arizona was scheduled to play the Minnesota Wild on Saturday and Sunday, and St. Louis was scheduled to play the Colorado Avalanche. Wild games are postponed through Feb. 9, and Avalanche games are postponed through Feb. 11 due to NHL COVID-19 protocol issues with each team. This game and on Monday were rescheduled from March 29 and 31. ... The Coyotes will play the Blues six straight games, the first time in NHL history a team will play an opponent that many times in a row during the regular season. ... Coyotes forward Drake Caggiula left the game in the second period with a lower-body injury. ... Blues forward Zach Sanford (flu) did not play. ... Blues center Tyler Bozak (upper body) was placed on injured reserve Saturday.