Mika Zibanejad scored twice, Mats Zuccarello scored and Georgiev made 29 saves for New York (32-32-8).
The Rangers trail the New Jersey Devils by 10 points for the second wild card from the East with 10 games to play.
"We didn't do a good job of kind of getting that forecheck going and getting some shots on net," Zibanejad said. "They're a good team and we talked about that before. I thought we worked ourselves back into the game, and it was unfortunate not to come up with the two points there."
Pietrangelo tied the game 3-3 with a power-play goal at 5:15 of the third period.
Nikita Soshnikov's first goal with the Blues gave them a 2-1 lead at 2:24 of the second period.
"It was a broken play going into the zone and [Chris Thorburn] dropped the puck for me," said Soshnikov, who played his eighth game with St. Louis after being acquired from the Toronto Maple Leafs on Feb. 15. "I saw the room for a shot and I shot it. I saw the room there and I went there."
Zuccarello tied it at 12:07 and Zibanejad's power-play goal at 14:34 gave the Rangers their first lead at 3-2.
Zibanejad tied the game 1-1 42 seconds into the second period.
Pietrangelo gave St. Louis a 1-0 lead at 15:49 of the first period, when the Blues outshot the Rangers 16-2.
"Well we followed the master plan in the first period, tire them out in our own end," Rangers coach Alain Vigneault said. "Our [defensemen] couldn't make a pass and our forwards couldn't get through the neutral zone, but after that, I liked our game. We played well in the second period, didn't give them much and we got some pretty good looks and were able to take a 3-2 lead."
