Stamkos, the Lightning captain, is the 95th player in NHL history to reach 1,000 points (494 goals, 506 assists), and the ninth active. He also extended his point streak to nine games (six goals, nine assists).
"I got the puck low in the corner, 'Killer' made a good play," Stamkos said. "I took a peek over my shoulder, I saw Cole coming down the backside and for a second I was just going to throw it back post, but then I didn't want to get yelled at in the meeting for turning the puck over. So, I thought I'd make the safe play behind the net, and obviously he made a great play out to Paul in front."
Andrei Vasilevskiy made 21 saves and had an assist, Cole had a goal and an assist, and Corey Perry had two assists for Tampa Bay (14-8-1).
Travis Konecny scored, and Hart made 23 saves for Philadelphia (8-11-5), which ended a 10-game slide (0-7-3) with a 3-1 win against the New York Islanders on Tuesday.
"There's not many positives from tonight, to be honest with you," Flyers defenseman Tony DeAngelo said. "Wasn't a good game all around. They outplayed us from start to finish."
Paul scored a power-play goal to give the Lightning a 1-0 lead at 16:02 of the first period on a one-timer from Perry.
Tampa Bay has scored with the man-advantage in nine straight games, an NHL high this season. It went 8-for-8 on power-play face-offs and scored once in three opportunities.
"The big thing for me is we're winning face-offs," Lightning coach Jon Cooper said. "When you win face-offs and you start with possession, the mojo starts kicking in."