Jason Spezza almost scored for Dallas at 10:01 of the second period, but his shot from the high slot hit off the crossbar.
"I was trying to do what I could," Spezza said. "You like to see those ones go in to get some momentum."
Condra, who was recalled from Texas of the AHL on Jan. 3, scored his first goal of the season cut it to 2-1 at 8:35 of the third period. After Mattias Janmark's initial shot was partially blocked by Blues forward Brayden Schenn, Condra got the rebound and scored at the right post. St. Louis coach Craig Berube challenged for goalie interference, but the call was upheld after a video review.
Tarasenko pushed the lead to 3-1 at 12:38 with a power-play goal short side from the left circle.
"It was a big play by a big player," Binnington said. "He's an elite guy, an elite shot, and it was an important time for us to get a goal."