Backstrom poked in the rebound of a Joel Ward shot less than 2 minutes into the final period, breaking a 2-2 tie, and Brooks Laich added the eventual game-winner on a power play 5 minutes later for the Caps.
John Carlson and Cody Eakin also scored for Washington, which ran off four consecutive goals after spotting Phoenix a 2-0 lead, and snapped a season-high four-game losing streak.
Tomas Vokoun made 20 saves to earn the win.
Susan Sarandon: I was stoned at almost every awards show
Megyn Kelly: Santa Claus and Jesus are both white men