Mager, 25, made his first ATP Tour semi-final by winning 7-6(4) 7-5 after leading 2-1 in the second set when play was halted on Friday.
The Italian held his opening three service games to serve for the match at 5-4. But after dropping serve to Austrian Thiem, Mager landed back-to-back winners and on his first match point, fired an ace out wide to complete the win.
Mager will play Hungary’s Attila Balazs in a semi-final later on Saturday. Balazs outlasted Spanish qualifier Pedro Martinez 2-6 6-4 6-2.
World number four Thiem was playing in his first tournament since reaching the Australian Open final last month.
(Reporting by Gene Cherry in Raleigh, North Carolina; Editing by Christian Radnedge)