"This is the type of performance we needed coming off a bit of a layoff," coach Davis Payne said. "We're healthy and ready to go."
Dennis Wideman scored for Florida and Tomas Vokoun stopped 26 shots.
The Blues scored the go-ahead goal at 16:00 of the third period. Florida turned the puck over at the blue line, giving St. Louis a 3-on-1 as it came down the ice. D'Agostini's wrister went high into the net to make it 2-1.
"We've been working hard but just not getting the outcome we wanted," D'Agostini said. "We keep working and getting pucks deep and eventually that's going to take a toll on their defense. I think we played a pretty solid game all around."
The Blues tied it at 1 with a minute left in the second.
The Panthers turned the puck over at their own blue line and Brad Boyes grabbed it and passed to Andy McDonald along the side boards. McDonald brought the puck through the left circle toward the net, then passed it to Backes in front, who easily beat Vokoun with a wrist shot for his 18th goal of the season.
Backes scored for the fourth time in seven games.
The Blues had lost 11 of 14 coming in.
"We finally put together a full 60 minutes and got rewarded for it," Backes said. "It took us a while but our goalie is good enough to keep us in games when we haven't been up to snuff."
In Florida's last 21 games, 18 have been decided by one goal.
"It's exhausting on everybody because you're playing 60-plus minutes right to the end," Panthers coach Peter DeBoer said. "We've got to find a way to win these one-goal games."
Florida took a 1-0 lead on a 5-on-3 power play in the first. Steven Weiss fed the puck to Wideman above the left circle and his one-timer beat Halak at 6:45.
The Panthers nearly went up by two goals at 8:11 of the second but a shot by David Booth hit the left post.
"We just didn't execute well," Vokoun said. "It's hard to score one goal and expect to win."
The Blues are hoping to build some momentum going forward.
"It's a long season. We need to make sure we go game by game," Halak said.
"We need to pick up points in every game if we want to make the playoffs."
Material from wire services and team media was used in this report.
1 - 0 FLA
1 - 1 Tie
2 - 1 STL
Delaying Game-Puck over glass
Holding the stick
Too many men/ice - bench