The contest was scoreless until Oduro received an accurate pass from Federico Higuain and beat TFC defender Doneil Henry to find the net.
The Crew broke a two-game winless streak with the result, improving to 4-4-3.
The Reds, meanwhile, dropped to 1-6-4 and still had trouble scoring despite the debuts of three new players -- Steven Caldwell, Jeremy Brockie and Bobby Convey.
Exploding whale video goes viral on Internet
Texas principal bans speaking Spanish, stirs controversy