"On this team we're focused on winning hockey games and that's it," Stamkos said. "I just want to chip in offensively whenever I can."
Boston, which has lost two straight after a three-game winning streak, got a goal from Michael Ryder. The Bruins (7-2) have dropped just two of nine road games this season.
"To win a hockey game you need to play 60 minutes and that's what we didn't do," Bruins defenseman Zdeno Chara said. "For the first 40 minutes Tampa was dominating the game and they deserved to win. It's too late when you turn on the engines in the last 20. That's not good enough."
The Lightning outshot Boston 36-27, including 31-15 through two periods.
Tampa Bay went up 2-0 early in the second when Tyrell (3:26) and Stamkos (5:15) scored 1 minute, 49 seconds apart. Stamkos was positioned in the low slot and redirected Martin St. Louis' backhand pass from the left circle past Boston goalie Tuukka Rask.
"I think we're playing with confidence and we've got a good thing going right now," Lightning center Nate Thompson said. "It feels good, but we can't accept that. We want more. We want to get better."
Purcell extended the Lightning lead to 3-0 with a rebound goal with 8:24 left in the second. The Bruins entered allowing just 1.8 goals per game.
Ryder ended Tampa Bay goalie Mike Smith's shutout bid with his 150th NHL goal at 10:38 of the third.
"We've done that a few times this year where it would take 40 minutes to get ourselves going," Julien said. "It catches up to you. Forty minutes of no show and 20 minutes of desperation."
Stamkos has reached 20 goals -- coming in 21 games -- during a season faster than any other Lightning player. Chris Kontos held the old mark, reaching 20 goals in the 27th game of the 1992-93 season.
A moment of silence was held for former NHL coach Pat Burns, who died Friday at the age of 58 due to cancer. Burns, a special assignment coach for New Jersey, had been a regular at Tampa Bay home games through the early part of last season. ... Jim Craig, the goalie for the 1980 gold medal-winning U.S. Olympic hockey team, attended the game. ... Tampa Bay LW Simon Gagne missed his 15th consecutive game because of a neck injury. Lightning coach Guy Boucher said there is a chance Gagne could return Wednesday. ... Bruins C Marc Savard (post-concussion symptoms) will undergo tests Tuesday that could clear him to take part in full-contact practices.