Lucic stepped in front of a shot by Cam Atkinson just inside the blue line and the puck took a big bounce toward the Columbus end with nobody between Lucic and Blue Jackets goalie Sergei Bobvrosky. Lucic corralled the puck and poked a shot between Bobvrosky's pads.
''It's my first overtime goal that I ever scored and it ends up being a big one,'' said Lucic, who leads the Bruins with eight goals.
Loui Eriksson and Shawn Thornton also scored the Bruins.
Boston goalie Chad Johnson stopped 32 shots, making his third start this season with No. 1 netminder Tuukka Rask getting a night off.
Columbus was coming off a 4-3 overtime loss in Washington on Tuesday and has won just once in the last eight games.
Boston's two goals in regulation both came on unintentional deflections and another fortunate bounce allowed the Bruins to extend their winning streak to four straight.
Atkinson's shot bounced hard off Lucic and headed straight in the opposite direction. Even Lucic called it a ''lucky bounce'' as the Bruins won despite getting outshot 12-7 in the third period and 2-1 in overtime.
''It feels the same. It's still an empty feeling,'' Columbus coach Todd Richards said of the second straight loss in OT. ''There were a lot of good things in the third period, but it's still the result. You know, we need to win games and we need two points here, not just one.''
The Blue Jackets also didn't help themselves, going scoreless on four power-play opportunities.
Bobvrosky made 28 saves for the Blue Jackets, who took a 1-0 lead in the first on Blake Comeau's second goal of the season. Boston scored twice before Nick Foligno restored the tie with a goal late in the second period.
The third was scoreless and the game appeared to be headed to a shootout before Lucic stepped in front of Atkinson's shot and skated in alone for the game-winner.
''He did a great job of blocking the shot and he got on his horse and took off,'' Boston coach Claude Julien said.
The Bruins closed a five-game homestand 4-0-1.
Thornton gave Boston a 2-1 lead on a slap shot from the circle that deflected off Columbus defenseman Jack Johnson's stick and beat Bobvrosky on the near post with 9:51 left in the second. It was the second time in the game the Bruins scored by just trying to get the puck on net.
Eriksson was credited with Boston's first goal without even taking a shot. He was facing the net when Zdeno Chara wristed the puck from the blue line and it deflected off Eriksson's skate and into the net to tie it a 1-1 with 1:50 left in the first period.
Comeau put Columbus up 1-0 with 7:12 left in the opening period on a shot from the slot off a backhand pass from Mark Letestu from behind the goal line.
Foligno tied it 2-all with 3:26 left in the second period when Johnson gave up a rebound straight to the slot. Foligno set up his own goal with a pass intended for Michael Chaput that and ended up reaching Johnson instead. Johnson made a save with his left pad, but the rebound went right to Foligno for a one-timer.
NOTES: Letestu played his 200th career game four years to the day after making his NHL debut, also against Boston. ... The Bruins avoided losing twice to Eastern Conference newcomer Columbus. The Blue Jackets beat Boston 3-1 on Oct. 12 in what remains Columbus' only win in the new Metropolitan Division. ... Both teams were opening a set of back-to-back games. The Blue Jackets are back at home Friday against Montreal and Boston visits Ottawa. ... Foligno had missed the last two games while attending to undisclosed personal matters.