Curtis Glencross and Anton Babchuk produced goals during the first 12 minutes of the middle period to put the Flames in front 4-2 but they could not get another shot past Jimmy Howard during regulation, overtime or the shootout.
Patrick Eaves had a power-play goal with 18 seconds left in the middle stanza to bring the Red Wings within one and defenseman Brian Rafalski's second goal of the game and the season tied it midway through the third.
Rafalski played in the 800th game of his career, which has spanned 11 seasons and two teams. He has reached double figures in goal scoring just twice in those 11 years.
Howard made 35 saves to pick up his 21st win of the season, tops in the NHL.
Miikka Kiprusoff took the loss while stopping 23 shots in Calgary's third straight setback.
Celebrity Families of 2014 [PHOTOS]