Elliott hadn't earned a win since April 27 against Chicago. His previous shutout was April 11 at Minnesota.
"It was good to get in there," said Elliott, who notched his 22nd NHL shutout. "Guys got me a lead pretty early. You kind of just settle in and start seeing the puck. It was a good outing."
Already ahead 1-0, the Blues broke the game open with three goals in just over five minutes in the second period.
St. Louis doubled its lead on Stewart's goal. Barret Jackman passed from the point to Stewart below the left circle for an angled shot from the side that got past Jacob Markstrom at 5:21.
The Blues stretched their lead to 3-0 with a power-play goal by Pietrangelo at 7:06. A drive by Backes was blocked by Markstrom, but Pietrangelo tipped in the rebound.
"To get that power-play goal really gave us a boost. The next shift we were able to find another one and put it in the back of the net," Backes said. "(Elliott) was up to the task. He was our best player when we needed him to be."
Steen finished the scoring when he wristed in a rebound at 10:42.
"I was just in the right spot at the right time. Good break," Steen said.
Steen didn't want to focus on his streak or his lead in goals.
"I think we've talked enough about that," Steen said. "I think Elliott is the reason we ended up winning the hockey game. He made some timely saves, some really big saves, and pretty much stood on his head at times."
Markstrom made 19 saves for Florida, which lost for the fifth time in six games.
"It's disappointing for all of us. Everybody feels bad about it," Panthers coach Kevin Dineen said. "The effort wasn't good enough to compete against one of the top teams in the NHL."
Dineen was clearly frustrated with his team's lack of success this season.
"If we can't get everybody to compete harder than we did tonight, then we'll just stumble along and get our wins here and there," Dineen said. "We need more participation in the hard parts of the game for us to have success.
"We're at the point now that it's not about skill for this group, it's about us trying to find the guys that are going to give us the best effort."
Roy put the Blues ahead with a backhander from the slot that hit the right post, bounced out, hit Markstrom's pads and drifted back across the goal line with 4:19 left in the first.
Blues C Maxim Lapierre returned after serving a five-game suspension for a hit on San Jose's Dan Boyle on Oct. 15. ... The Blues played an NHL-low 10 games in October but will play 15 in November. ... The Panthers have lost four straight to the Blues, including a 7-0 rout on Oct. 5. ... Florida went 1-3-2 on a season-high, six game homestand.
Copyright by STATS LLC and The Associated Press
SUNRISE, FL - NOVEMBER 1: Alexander Steen #20 of the St. Louis Blues skates prior to the game against the Florida...