Berglund and Backes scored within 2:01 of each other early in the third to break a 1-1 tie and the Blues won for the seventh time in their last eight games.
Backes' goal was his second of the night. He had tied the score on a power play at 11:38 of the second period.
Jaroslav Halak made 24 saves for the Blues, who lead the Central Division by four points over Detroit.
Jonas Hiller gave up three goals on 27 shots for the Ducks, who have lost four of their last six.
Corey Perry scored Anaheim's only goal to open the scoring at 4:18 of the second.