Oshie scored 3:29 into overtime, jamming the puck in from the side of the net as the Blues won 3-2 on Thursday night.
"It was a pretty lucky bounce," Oshie said. "It hit off me and went straight toward the net and I just started whacking and poked it in there. We'll take the point for sure."
Boston goalie Tuukka Rask didn't have much a chance on Oshie's unassisted goal.
"It hit Oshie's skate and it hit the back of the net and it was on his stick and then he backed it off my pad I guess," Rask said. "What are you going to do?"
It was the third straight home game that went to overtime for the Blues.
Alexander Steen and Jaden Schwartz also scored for St. Louis, which blew a 2-0 lead in the third period as Boston scored in a span of just over 2 minutes. St. Louis also blew a two-goal lead Tuesday in a shootout loss to Ottawa.
Boston had won in its last three trips to St. Louis, and this was the first Blues' home win against the Bruins since Oct. 6, 2006, in a shootout. The last regulation win at home against Boston came on Dec. 18, 1999.
"I think when you play Boston, I think it's a great evaluator," St. Louis coach Ken Hitchcock said. "It gives you a real good read on your competitive level. It gives you a great read on your execution under pressure."
Steen gave the Blues a 1-0 lead with his 28th goal when he ripped a slap shot from the top of the right circle to beat Rask at 15:48 of the first period. Jaden Schwartz fed the puck to Steen, who has eight points in his last nine games.
Schwartz made it 2-0 when he snapped a wrist shot from the left circle at 3:32 of the second period. The shot by Schwartz grazed the left shoulder of Rusk, who was screened by Kevan Miller. It was the 18th goal by Schwartz.
St. Louis' Jaroslav Halak stopped 24 shots in the first two periods before allowing two on the first seven shots in the third.
"Our most competitive player tonight was our goalie," Hitchcock said. "I mean, the goalie was outstanding. He helped us a lot."
Krejci scored for Boston on a wrist shot from the top of the slot at 9:16, and Marchand then scored on a rebound of a shot by Johnny Boychuck at 11:24.
"I don't think I could have done anything different about those two," Halak said. "I think we responded the right way after they scored. We skated hard. We battled hard."
The Blues went 0-for-3 on the power play, and are now 0-for-12 in the last three games.
Boston is 7-2-1 in its last 10 games, and the Bruins dropped their first on the road in their last six games.
"I thought our team played extremely well and probably deserved better," Boston coach Claude Julien said. "It was an unfortunate bounce to lose that game in overtime, but still a great effort from our team."
Hall of Fame inductee and former St. Louis Rams player Aeneas Williams dropped the ceremonial first puck. ... Boston captain Zdeno Chara was not at the game. He has left for the Olympics, where he is to carry Slovakia's flag in the opening ceremonies. Defenseman David Warsofsky took his place in the lineup. ... Blues F Vladimir Sobotak (knee) and F Vladimir Tarasenko (flu) did not play. ... Boston D Adam McQuaid (lower body) did not play. ... Blues F Brenden Morrow played his 900th career game. He is the 47th active player to accomplish the feat.