David Krejci scored the tying goal with 2:33 remaining in the third period, and Halak made 32 saves for Boston, the No. 4 seed, which lost four straight after winning Game 1 3-2.
"This team is resilient," Bruins coach Bruce Cassidy said. "We're disappointed. We thought we were the better team tonight. We wanted to play on. Put ourselves in a hole, obviously, but felt we played well enough to win tonight, get it to Game 6, and then see what happens from there."
The Bruins, who were 5-8-0 in the postseason, won the Presidents' Trophy with a .714 points percentage (44-14-12) in the regular season after advancing to the Stanley Cup Final last season, which they lost to the St. Louis Blues in seven games.
"It's a very disappointing finish. We had a great year," Boston forward Brad Marchand said. "We expected better out of this year. We thought we had a chance. I mean, Tampa's a great team, don't get me wrong. I don't want to take anything away from them. They're a great team.
"The way things were rolling throughout the season, we thought we were going to go all the way. … We walk away without anything to show for it. So it's tough."