The visitors built up a 3-0 lead en route to a 4-2 victory.
It was Calgary's fourth straight loss, pointing them at 15-7-6.
The Flames now head out for a two-game U.S. road swing before returning home to play six of their final seven games of 2021 at home.
Tkachuk, who turned 24, scored, as did Sean Monahan, while Jacob Markstrom made 23 saves.
Lucic collected an assist on Monahan's tally.
The Flames held a significant edge in shots, with Linus Ullmark much busier while making 40 stops.
Calgary generated a lot of looks in the first period, finishing with a 15-7 edge in shots, with a 14-5 differential in scoring chances.
But they trailed 1-0 after 20.
Johnny Gaudreau was doing what he does best early in the frame - creating pressure in the offensive zone and setting up teammates for Grade-A chances - and one such possession led to Boston's Brad Marchand getting whistled for a hook on Tkachuk and giving the homeside their first powerplay.
It was shortlived, though. Just 17 seconds into the advantage, Monahan was called on a slash to create some four-on-four action.
Despite controlling the play, it was the Bruins who got on the board first late in the frame at 17:43 when Connor Clifton's shot from the right wing on a rush squeaked through Markstrom.
The Flames second man-up of the night came 1:55 into the middle stanza, and they got cracks at Ullmark via an Andrew Mangiapane shot, a one-handed rebound attempt by Lucic and a point-blank slot shot from Elias Lindholm. However, the Bruins goalie stopped all three.
Boston made it 2-0 at 5:22 when Marchand - with his back to Markstrom some 10-feet away - made a no-look tip of David Pastrnak's point shot that found twine.
Calgary made good on its third powerplay of the night.
Tkachuk - parked beside the blue paint - shovelled the puck home after a shot from Rasmus Andersson at 12:19. Gaudreau also picked up a helper.
