Geekie has scored eight goals in his past six games and has 20 on the season, which is tied with Nathan MacKinnon of the Colorado Avalanche for the NHL lead.
Jeremy Swayman made 24 saves and stopped all three attempts he faced in the shootout for the Bruins (15-12-0), who had lost four of six, including 6-2 to the New York Rangers on Friday. Elias Lindholm had two assists.
Lucas Raymond and Michael Rasmussen scored, and Dylan Larkin had two assists for the Red Wings (13-11-2), who have lost four in a row (0-3-1). Cam Talbot made 17 saves.
Geekie gave the Bruins a 1-0 lead at 4:25 of the second period. Red Wing defenseman Ben Chiarot tried to clear the puck up the right boards, but it went right to Lindholm, who sent a quick one-timer toward the net, where Geekie redirected it short side on Talbot.
Raymond tied it 1-1 at 6:38 of the third period, chipping in a slap pass from Larkin at the left post.
Geekie put the Bruins back in front 2-1 at 13:39, scoring a power-play goal with a one-timer from the left circle.
Boston was 1-for-2 on the power play. Detroit was 0-for-5.
Rasmussen tied it 2-2 at 18:06, tapping in a centering pass from Patrick Kane with Talbot pulled for the extra skater. The goal came shortly after Alex Steeves missed on his backhand attempt at the empty net.