The Ohio outfit put together a strong showing against a Chicago side that played the majority of the game a man down.
Oduro's opener came in the 15th minute, when he picked a ball up on the right side of the Fire box unmarked and rocketed home his shot to put his team in front early on.
Bakary Soumare's rash tackle on Jairo Arietta 14 minutes later made life even more difficult for the visiting side. The big defender was handed his marching orders, and the Fire would have to face over an hour of game time with 10 men.
Higuaín netted his goal from the spot in the 70th minute, wrapping up the win after Arietta earned a penalty.
Anor tacked on the third goal in the 76th, netting his third of the season by finishing off a clinical counterattack from the Crew.