Artturi Lehkonen scored, and Scott Wedgewood made 12 saves for the Avalanche (28-20-2), who have lost three straight (0-2-1).
"I thought our guys did a nice job," Colorado coach Jared Bednar said of playing less than 24 hours after Rantanen was traded following 10 years with the team. "They came ready to play and play hard and play well. They're all pros. I'm sure their minds thinking of that kind of stuff last night and their teammate prior to the game at points. But I thought they did a great job with their focus. I thought we played hard, played to the game plan, didn't capitalize on some of our chances and we kind of fell asleep for a couple minutes and it cost us the game, but I thought they did a nice job with their focus."
Lehkonen gave the Avalanche a 1-0 lead at 1:30 of the second period. Juuso Parssinen controlled the puck, gained the offensive line, and fed a backhand pass to Lehkonen. He received it in the right face-off circle and his shot from the dot beat Swayman to the stick side.
The Bruins, who had eight shots on goal through two periods, tied it 1-1 when Geekie scored 25 seconds into the third period.
Geekie then gave Boston a 2-1 lead at 5:44. Pastrnak controlled the puck behind the net, skated around Wedgewood's right side and found Geekie at the far post for the backdoor tap-in goal.
Geekie has six goals in his past seven games.
"It's growing," he said of his chemistry with Pastrnak and Pavel Zacha. "We've played together for a while now, and it's always easier when you learn the habits of the guys you're playing with and it's always fun to play with guys with lots of skill. For me, you're in awe with the things [Pastrnak] does with the puck and I just try to get open. It's lots of fun right now, so let's keep having fun."