Alex Killorn had a goal and two assists and Steven Stamkos also scored for the Lightning, who have won all five games this season against West teams. Tampa Bay, which has made the playoffs once over the previous six seasons, is off to a 10-4-0 start.
"I think progress is probably the correct word because we've played some of these upper echelon teams," Tampa Bay coach Jon Cooper said. "I thought we earned the victory against an extremely good opponent. That's the progress because we're playing an elite team and I think we more than hung with them."
Two of Tampa Bay's out-of-conference victories have come against the defending Stanley Cup champion Chicago Blackhawks.
"Are we the perfect hockey team? Not even close," Cooper said. "But are we a team that doesn't quit and responds to challenges? They do."
Connolly, playing in his second game since being recalled from Syracuse of the AHL, put Tampa Bay ahead 3-2 when he redirected Radko Gudas' shot at 8:37 of the third. Filppula made it 4-2 with 3:59 remaining.
Tampa Bay's Ben Bishop turned aside a shot by Chris Stewart during a late first-period breakaway. He finished with 28 saves in his first game against the team that drafted him.
"I thought it was a little bit of a measuring stick," Bishop said of beating the Blues, who are 8-2-2. "We want to be in the same category as those guys."
"We need to reset the dial and get more participants," Blues coach Ken Hitchcock said. "This is a real good evaluation, because we've had opponents where we've played kind of average and won, and now we had to play really good and we were still just above average."
Stamkos scored his 11th goal of the season to give Tampa Bay a 2-1 lead 1:43 into the second during a 2-on-1. The center, who has a goal in four consecutive games, briefly dragged the puck before putting an in-close shot past Jaroslav Halak.
"We cannot find any excuses," Halak said. "They just outplayed us."
Halak stopped 21 shots, including nice saves on Stamkos and Martin St. Louis while the Blues were short-handed in the second.
Schwartz got St. Louis even at 2-2 from the slot 2:02 after Stamkos' goal.
Pietrangelo extended his goal-scoring streak to four games and put the Blues up 1-0 at 7:50 of the first.
Killorn tied it at 1-all with 9:16 left in the first. It was the first even-strength goal allowed by St. Louis in 225 minutes, 55 seconds.
St. Louis' Alexander Steen, who picked up an assist, had his six-game goal streak end. He scored eight of his 12 goals this season during the stretch. ... Stamkos has six goals and 10 points during a six-game points streak. ... Lightning D Mark Barberio assisted on Stamkos' goal for his first NHL point. ... Blues D Vladimir Sobotka was minus-3 in the game.
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TAMPA, FL - NOVEMBER 2: Goalie Ben Bishop #30 of the Tampa Bay Lightning faces shots from teammates during the pre...