Alex Killorn and Mathieu Joseph scored for the Lightning (29-10-5), who have won six of their past seven (6-1-0). Andrei Vasilevskiy made 31 saves.
Damon Severson had a goal and an assist for the Devils (15-22-5), who have lost three straight. Jon Gillies made 32 saves.
"They're the best team in the League until someone knocks them off," Devils forward Jack Hughes said of the two-time defending Stanley Cup champions. "They know how to win. We'll get to that point one day. That's a better hockey team than us, but we played a really good game tonight and it's disappointing we didn't get two points."
Severson gave New Jersey a 1-0 lead with a power-play goal at 4:33 of the first period. His shot from the point got through traffic and beat Vasilevskiy on the stick side.
Killorn's power-play goal tied it 1-1 at 1:46 of the second period. He gathered a rebound in front and slipped the puck past Gillies.
"If you're not going to go to those areas to score and compete in those areas, you're not going to win," Tampa Bay coach Jon Cooper said. "You get into these one-goal games, you're going to need those, and we had just enough tonight."
After Lightning defenseman Jan Rutta was assessed a four-minute penalty for high-sticking at 12:21, the Devils struggled to keep the puck in the zone during the power play. They gave up a breakaway to Cirelli, who was awarded a penalty shot at 13:50 when Severson was called for hooking.
Gillies stopped Cirelli on the penalty shot, but Joseph put the Lightning ahead 2-1 with a shorthanded goal at 14:16, scoring on his own rebound.
"We did exactly what we didn't need to do," Hughes said. "Obviously we gave up the penalty shot and we gave up the goal too. It's not very good out of us, which is disappointing because we've been really good on the power play as of late. That was a chance for us to make a huge difference."
Joseph said, "Sometimes their power play is a little on its heels defensively. Fortunate bounce. Good save at first, but the puck was right there. I'll take it, but good bounce for sure."