After Pat Maroon put a wraparound shot on net, the puck went through the crease for Palat to score on a wrist shot near the right post.
"Even when they scored, we scored," Palat said. "We responded every time. I thought we played really well. I think in the third, we stopped skating a little bit. Other than that, I thought we played well."
WATCH: [All Bruins vs. Lightning highlights]
Brad Marchand tied the game 3-3 when he scored his second goal of the night for Boston with 3:58 left in the third period.
"I thought we had good looks," Bruins coach Bruce Cassidy said. "We just didn't have the volume of attempts. We had point-blanks in the slot. … There were some good looks for us. There just wasn't the quantity of the kind of play like our tying goal by [Marchand]. So we certainly need a little bit more of that."
Blake Coleman scored twice, and Andrei Vasilevskiy made 22 saves for Tampa Bay, the No. 2 seed in the East, which is 4-0-0 in overtime this postseason. The Lighting trailed by one goal twice.
"If you want to have any chance of winning and going deep in the playoffs, adversity strikes in the weirdest ways and you never know when it's going to happen," Lightning coach Jon Cooper said. "It's like a sandwich that doesn't taste very good, if you know what I mean? … If you have mental weakness at all, you're probably sitting there saying, 'Poor us.' It's just not in this group. That's not how they operate."