The Capitals trailed 4-1 early in the third period but began their comeback midway through the period when Mike Green scored with 8:40 remaining.
Dmitry Orlov cut the deficit to 4-3 with a long slap shot from the point 5 minutes later and Alex Ovechkin scored his 27th of the season with just 48 seconds to go, forcing extra time.
During the shootout, Backstrom fired in a wrist shot to the blocker side of Steve Mason in the third round and Philipp Grubauer then turned aside Sean Couturier to secure the Capitals' fourth win in five games.
Ovechkin and Marcus Johansson each got a goal with an assist and Grubauer made 24 saves.
Claude Giroux had a goal and an assist for Philadelphia.
Couturier, Mark Streit and Jakub Voracek each scored and Michael Raffl contributed three assists.
Mason ended with 29 stops.