Stars forward Jamie Benn was called for tripping Tyler Johnson at 5:10 of overtime.
"It's playoffs, overtime, you expect 5-on-5, you expect to battle it out," Dallas forward Joe Pavelski said. "We got a call against, it's unfortunate, but it doesn't change our mindset. We thought there was a good chance this would be a long series. We have to do a job and extend it now."
Pavelski scored two goals, Tyler Seguin had two assists, and Anton Khudobin made 30 saves for the Stars, who had a 2-0 lead in the first period.
"We would've won if we would've gotten everything out of everybody, I think," Seguin said. "We've got a quick turnaround. I believe in this team, believe in the boys. We have another level here, so the quick turnaround's going to be nice."
Teams that win Game 4 to take a 3-1 lead in a best-of-7 Cup Final (since 1939) are 27-0 winning the series.
"(After the game) it was laser focus," Lightning coach Jon Cooper said. "They came here to do something and they're on their way, but like we've said all along, we haven't won anything yet, and they truly believe that. They know what it takes to win a series. We've still got to win one more, and we're not taking anything for granted."
Pavelski tied the game 4-4 at 11:35 of the third period when his shot deflected in off Shattenkirk. It was his 60th NHL postseason goal to tie Joe Mullen for most by a United States-born player.