The Tigers were clinging to a 3-point lead with 13 seconds left as the Cowboys played for a possible score-tying basket in the title game of the Old Spice Classic.
But Goodwin tipped a pass by Marcus Smart and the ball went out of bounds off OSU's Phil Forte with 2.2 seconds remaining for a turnover.
Joe Jackson went 4-for-4 from the free-throw line in the closing seconds to secure the win for Memphis (5-1).
Goodwin also provided eight rebounds and Chris Crawford scored 16 points.
Forte put in 19 points for the Cowboys (7-1), who lost for the first time this season.
Smart, a sophomore sensation averaging more than 21 points a game, was held to a season-low 12 points on 4-of-13 shooting.