Phoenix was booed by the small hometown crowd late in a lackadaisical first period and was again sluggish to start the second. The Coyotes righted themselves after a timeout by coach Dave Tippett midway through the second, getting goals from Ray Whitney and Korpikoski.
Radim Vrbata had assists on both goals, helping Phoenix move within a game of San Jose in the Pacific Division and win 40 games in consecutive seasons for the first time in franchise history.
Kevin Shattenkirk scored for the Blues in their fourth loss in five games.
Phoenix was coming off a frustrating loss Sunday.
Despite creating few good scoring chances against Chicago, the Coyotes were still tied late into the third period. The Blackhawks won it on Chris Campoli's power-play goal with just 6 minutes left after a holding call on Phoenix captain Shane Doan that the Coyotes didn't agree with.
Before that, though, Phoenix had been on a roll, scoring at least three goals in seven straight games while earning a point in each (6-0-1). That put the Coyotes within reach of the Sharks in the Pacific with two weeks left.
With their lopsided loss to the Sharks on Saturday, the Blues had dropped 12 points out of the Western Conference's final playoff spot with 10 games left. Worse yet, St. Louis has five teams to pass to get to that eighth spot.
Early on, Phoenix played like the team out of the playoffs, not the Blues.
The Coyotes looked scattered in the first period, laboring through big chunks of time in their own end and hearing boos from the home crowd while spending most of their two power plays chasing pucks the Blues had cleared to the other end.
Shattenkirk put the Blues up midway through, one-timing a pass from Chris Stewart, who gathered a long pass from Ian Cole at the side of the crease after a long shift in Phoenix's zone.
It didn't get much better for Phoenix to start the second period. The Blues camped out in the Coyotes' zone for most of the opening 6 minutes, leading Tippett to call a timeout to give them a few choice words.
Whatever he said, it worked. The Coyotes got their first shot of the period about 2 minutes later and Whitney scored his 16th goal of the season right after that, one-timing a mis-hit shot by Korpikoski to tie the game.
Korpikoski put Phoenix ahead in the closing minutes of the period, sending a wrist shot past Jaroslov Halak's glove side for his 18th of the season.
Bryzgalov made the lead stand by surviving a flurry of shots in the closing seconds to move within three wins of Bob Essensa's franchise record of 129 career victories.
Notes: Coyotes LW Paul Bissonette was involved in two fights, early in each of the first two period. He and Blues RW Ryan Reaves dropped the gloves 1:58 into the game and Tyson Strachan got into it with Phoenix's enforcer 2½ minutes into the second. ... St. Louis is 2-20-2 when trailing after two periods.