Frolik stole the puck from Brad Richardson after a faceoff in the Flames end and skated in at top speed and ripped a shot inside the post on Coyotes goalie Mike Smith.
Matthew Tkachuk scored for Calgary (7-10-1) in regulation time. The Flames snapped a three-game home losing skid.
Tobias Rieder scored for Arizona (5-9-1). The Coyotes wrap up the short two-game trip against Vancouver on Thursday.
Down 1-0 and held to only 13 shots through 40 minutes, the Flames tied it 6:40 into the third. From a sharp angle just above the goal line, Tkachuk ripped a shot under the crossbar on Smith. It was the rookie’s fourth goal.
The third period saw Calgary controlling play but unable to get the go-ahead goal. Among Smith’s stops in the period was a stick save off Dennis Wideman and a sharp pad stop off Troy Brouwer.
After an uneventful and scoreless first period, Calgary went to the attack to begin the second. On the opening shift, Tkachuk shot wide on a partial breakaway, Smith jabbed out a pad to deny Frolik from in-tight and TJ Brodie’s wrist shot from the low slot was turned aside.
Five minutes later, Frolik had another dangerous chance from in front only to be thwarted again, this time Smith throwing up his glove to stab Frolik’s snapshot from 20 feet out.
Arizona took the lead with 18 seconds left in the second when Rieder pounced on a Justin Martinook rebound and buried it behind Flames starter Chad Johnson.
It was the first game in what will be an extended stretch for Calgary without leading scorer Johnny Gaudreau, who underwent finger surgery on Wednesday after getting injured Tuesday night in Minnesota.
Sam Bennett took his spot on the top line with Sean Monahan and Brouwer.
In his return from missing 12 games with a knee injury, Smith had 22 stops. His record falls to 1-0-1.
After not playing for nine days, Johnson got the start again, 24 hours after a 27-shot shutout in Minnesota. He made 20 stops to improve to 4-2-1.
Notes: Linden Vey was inserted into the forward spot opened up by Gaudreau… Arizona’s Martin Hanzal (lower body) returned after missing six games…. Defencemen Luke Schenn and Kevin Connauton returned to the Coyotes line-up with Connor Murphy and rookie Jakob Chychrun taking a seat.