Kostitsyn drew Montreal within 4-3 just 10 seconds after Alfredsson's third goal later when he scored at 14:25, and Streit brought the Canadiens even at 16:26 when he beat Martin Gerber with a slap shot from the point.
Carey Price, Montreal's highly-touted goaltending prospect, replaced Cristobal Huet at the beginning of the third and made eight saves in his final bid to make the opening night roster. The Canadiens open in Carolina on Wednesday.
Price stopped Chris Kelly's backhand 2:28 into overtime and made a fine pad save on Alfredsson's point shot with 44 seconds remaining with the Canadiens short-handed after Mikhail Grabovsky was sent off for hooking at 3:29.
Eaves gave Price little chance on the winner when he redirected Alfredsson's pass across the goalmouth inside the right post from the edge of the crease.
Gerber, who appeared in all six preseason games and will start the season opener Wednesday in Toronto, made 35 saves in his fifth preseason start after the Senators decided to allow Ray Emery more time to recover from offseason wrist surgery.
Emery stopped seven of nine shots while playing the first two periods of Tuesday's 4-2 win over Philadelphia before Gerber replaced him for the third.
Ottawa recalled goalie Brian Elliott from Binghamtom of the AHL earlier in the day.
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Arena: Canadian Tire Centre Location: Ottawa, Ontario