Corey Perry opened the scoring with his 34th goal of the season at 4:18 of the second period, assisted by Matt Beleskey and Ryan Getzlaf. But Backes tied the score with a power play goal at 11:38 of the second period.
Patrik Berglund scored the go-ahead goal at 4:47 of the third period, his 14th of the season, and two minutes later Backes scored his second goal of the night.
Jonas Hiller saved 24 of 27 shots in defeat for Anaheim. Jaroslav Halak saved 24 of 25 shots for St. Louis, who improved to a whopping 28-4-4 at home this season.
The Ducks continue their road trip on Saturday night against the Colorado Avalanche at the Pepsi Center in Denver.
For more news and notes regarding Ducks hockey, be sure to read Anaheim Calling.