The victory kept the Flames within two points of the Detroit Red Wings for the final Western Conference playoff berth.
Bourque's unassisted breakaway goal late in the second period gave Calgary a 2-1 lead and proved to be the game-winner. Miikka Kiprusoff made 22 saves for Calgary, which has six of its nine remaining regular-season games on the road.
Scott Niedermayer scored the Anaheim goal. Jonas Hiller recorded 25 saves.
Dawes opened the scoring with a deflection goal on a power play at 15:11 of the first period. Niedermayer tied the contest at 14:27 of the second period but Bourque put Calgary back in the lead for good at 18:25.
Hagman closed out the scoring with his 22nd goal of the season at 3:05 of the third.