Barton also had 14 rebounds to lead the Tigers (5-2) to their third straight win.
Memphis led 27-26 at halftime but took control after the break. The Hurricanes cut the Tigers' lead to 59-51 with 4:19 to play but didn't score again until 1:34 remained, and Memphis padded its lead with five free throws by Joe Jackson and two by Barton.
Chris Crawford and Tarik Black scored nine points each for Memphis.
Miami, which has lost three of its last four, was led by Kenny Kadji, who scored 12 points and grabbed six rebounds. Durand Scott and Trey McKinney Jones added 10 apiece.