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.
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