The Bruins, on an 0-for-13 power-play drought coming in, scored three power-play goals, two of them in 5-on-3 situations. Savard had all three of his points on the power play.
Boston had leads of 3-0 and 5-2, but a two-goal rally by Atlanta, the second goal by Ilya Kovalchuk with 4:52 left, made it close. Patrice Bergeron hit the empty net with 3.5 seconds left.
The win, the Bruins' second straight since losing four in a row, was Boston's eighth in a row over the Thrashers, who haven't beaten the Bruins since Dec. 29, 2007.
The Thrashers, opening a seven-game road swing (two of the games in Boston), have lost three straight games.
Savard scored to start the uprising at 9:44 of the period before Paille and Sobotka connected 20 seconds apart to chase goalie Ondrej Pavelec (three goals on eight shots). The Thrashers rallied on goals by Bryan Little and Maxin Afinogenov (one goal, two assists) in a 3-2 first period, but Zdeno Chara, who set up Savard's goal, delivered during a two-man advantage early in the second period, off Savard's pass - and Savard then assisted on Michael Ryder's 5-on-3 score in the third period.
Evander Kane scored with 6:02 left for Atlanta, and Kovalchuk then connected.
Dennis Wideman added two assists for the Bruins.
Savard, stationed in the circle to the right of Pavelec, took a pass from Chara, waited for the goalie to drop down and fired the puck high into the net for his first goal in seven games.
Paille and Sobotka - playing in his 100th NHL game - then both scored on short tap-ins to KO Pavelec in favor of Johan Hedberg. Mark Recchi set up Paille's fourth of the year and Shawn Thornton did the same for Sobotka's fourth.
The offensive-minded Thrashers came right back as Little and Afinogenov scored 2:44 apart, Afinogenov assisting on Little's slapper that found its way through Tim Thomas - while Kovalchuk set up Afinogenov with a nifty pass - Kovalchuk extending his points streak to eight straight games.
Chara gave the Bruins a 4-2 lead on the 5-on-3 at 2:14 of the second period. Atlanta thought it had tied the game in a goal-mouth scramble but Colby Armstrong was called for goalie interference, negating the goal and putting the Bruins on the power play. Fifteen seconds later, Marty Reasoner went off for tripping, setting up the lengthy 5-on-3, and Chara scored his third of the year.
NOTES: The Thrashers will be back in Boston next Wednesday night. ... Bruins coach Claude Julien ran his 200th game behind the Boston bench. He's 112-59-29. ... Thornton had gone 14 straight games without a point, but has three assists in the last five. ... Kovalchuk has five goals and seven assists during his streak. ... Bruins left wing Milan Lucic, out since Nov. 25 with a high ankle sprain, is skating and hopes to play as early as the Jan. 1 Winter Classic at Fenway Park, if not soon after.