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.
|David Backes (1) Backhand shot - ASST: Andy McDonald (1), Jamie Langenbrunner (1)|
1 - 0 STL
|Joakim Lindstrom (1) Snap shot - ASST: Daniel Winnik (1), Ryan Wilson (1)|
1 - 1 Tie
|PPG - T.J. Oshie (1) Snap shot - ASST: Andy McDonald (2), Alex Pietrangelo (1)|
2 - 1 STL
|Patrik Berglund (1) Snap shot - ASST: T.J. Oshie (1), Barret Jackman (1)|
3 - 1 STL
|Carlo Colaiacovo Tripping - 2 min against Gabriel Landeskog|
|Patrick Bordeleau Kneeing - 2 min against T.J. Oshie|
|Ryan Reaves Fighting - 5 min against Patrick Bordeleau|
|Patrick Bordeleau Fighting - 5 min against Ryan Reaves|
|Ryan Reaves Boarding - 2 min against Kyle Quincey|
|Ryan Wilson Holding - 2 min against Andy McDonald|
|Stefan Elliott Hooking - 2 min against Chris Stewart|
|Kevin Porter Too many men on the ice - 2 min|
|B.J. Crombeen Unsportsmanlike conduct - 2 min against Cody McLeod|
|Cody McLeod Unsportsmanlike conduct - 2 min against B.J. Crombeen|