Oduro, who entered the match in the 65th minute, tied the score in the 83rd and added his second goal just seconds into stoppage time.
FC Dallas (3-3-3), which fired Coach Steve Morrow Tuesday, broke a four-game winless streak, with Marco Ferruzzi making his debut as interim coach. The Hoops had not won a Major League Soccer game in more than a month.
Real Salt Lake (2-5-2) lost its second straight. Fabian Espindola scored Salt Lake's only goal in the 69th minute.