The Lightning started strong with Cirelli capping a furious opening flurry to give them a 1-0 lead 36 seconds into the game. Kuemper made saves on Ryan McDonagh, Alex Killorn and Brandon Hagel before Erik Cernak's shot from the right circle knocked off his mask and Cirelli put the rebound in for his second goal in as many games.
Though Tampa Bay dominated the first period, outshooting Colorado 17-4, it could not add to its lead against Kuemper, who rebounded from being pulled in the second period of a 6-2 loss in Game 3, when he gave up five goals on 22 shots.
"I thought he did a great job for us," Avalanche coach Jared Bednar said. "That was a hard-fought hockey game. Lots of talk about Vasilevskiy and rightly so, but 'Kemps' was equally good tonight."
MacKinnon tied it 1-1 with a power-play goal 5:17 into the second. He went to the net after passing the puck to Mikko Rantanen in the right circle, and Rantanen's return pass deflected in off MacKinnon's skate in front.
But Hedman put the Lightning back in front 2-1 at 10:42 by finishing off a weaving rush with a backhand from the right circle that went in over Kuemper's right pad.
Cogliano tied it 2-2 at 2:53 of the third period by when Nico Sturm's rebound backhand deflected in off his leg in front.
That set the stage for Kadri to be the hero in overtime. Kuemper started the play by passing the puck ahead to Artturi Lehkonen, and Lehkonen fed Kadri at the offensive blue line, catching the Lightning in the middle of a change.
"It was an extended shift in our zone," Sergachev said. "I couldn't gap up and [Kadri] flew in with a lot of speed, put it under my stick and just put it in the chicken wing on 'Vasy.'"