Tulane trailed by a point with a minute to play when Osetkowski struck from the paint to give the Green Wave (5-3) the lead, 63-62, with 33 seconds left. New Orleans didn't score after that.
Tulane led 33-24 at the break but its advantage eroded in the second half and New Orleans went ahead 62-61 on a 3-pointer by Christavious Gill with a minute to play.
Gill led the Privateers (2-4) with 15 points. Nate Frye added 11.