Joakim Lindstrom scored for Colorado, which has dropped three of four.
Oshie beat Giguere when McDonald's shot came off the back boards to the other side of the net 9 minutes into the third period.
Berglund scored 3 1/2 minutes later on a shot through Giguere's pads from the left circle for a two-goal advantage.
Backes gave St. Louis a 1-0 lead he got McDonald's backhand shot in the slot and sent a rebound of his own past Giguere with 2:32 left in the second period.
Lindstrom tied it 1:42 into the third period on a one-timer from the right circle, beating Bishop, who seemed to lose sight of the puck as it went through the crease to the far side.