Kucherov gave the Lightning a 1-0 lead when he scored on a breakaway at 5:33 of the first period off a turnover by Heiskanen as he was being pressured by Palat.
"You don't see him make those mistakes, but he didn't get a whole lot of help there either," Stars coach Rick Bowness said. "Sometimes you just need a little help to make a simple play."
Stamkos put the Lightning ahead by two at 6:58 when he scored with a high shot to the far side from the bottom of the right face-off circle.
Stamkos didn't score in the 2015 Stanley Cup Final, a six-game loss to the Chicago Blackhawks, when he played 121:12 in 148 shifts. It took him 1:26 of ice time to score his first goal in this Cup Final.
"At this time of the year, you want to do anything you can to help your team win," Stamkos said. "I've watched these guys be so committed to what our end goal is, and to be part of it tonight, it was a dream come true."
Dickinson scored shorthanded at 11:19 to make it 2-1.
Tampa Bay took a four-goal lead in the second period.
Hedman scored on the power play at 54 seconds to make it 3-1. It was his 10th goal, third behind Brian Leetch (11 for the New York Rangers in 1994) and Paul Coffey (12 for the Edmonton Oilers in 1985) for most by a defenseman in a single postseason.
"That's a huge 3-1 goal for us after we'd given up a shorty just earlier in the last period," Cooper said. "The timing of a lot of things he's done has been remarkable, but I guess if you do great things all game, your timing is always going to be good."