The Red Wings lost their fifth game in a row. Six of their 11 setbacks this season have come in overtime or a shootout.
Detroit took a 3-1 lead into the third period, but the Capitals rallied on scores by Alex Ovechkin and Michael Latta. The tying goal came with 8:06 to go in regulation.
Backstrom's goal was the only one in the shootout and came against Jimmy Howard, who stopped 30 shots.
Braden Holtby made 34 saves before stopping all three attempts in the shootout.
Johan Franzen scored two of Detroit's goals, both coming in the first period. Danny DeKeyser gave the Red Wings a two-goal lead with the only score of the second period.