Federer, a 17-time Grand Slam winner, made 34 unforced errors in the Rome encounter ahead of the French Open, which starts on 22 May.
Austrian Thiem, 22, will face the winner of Richard Gasquet against Kei Nishikori in the quarter-finals.
Other third round matches on Thursday include Britain`s Andy Murray playing Frenchman Jeremy Chardy, while Australian Nick Kyrgios takes on seven-time Rome champion Rafael Nadal.
Federer`s defeat means Murray will return to the world number two ranking next week.
