Defending champion Novak Djokovic had an unexpected stumble at the start of his match, losing the first set to Thomaz Bellucci to love, before recovering to beat the Brazilian 0-6 6-3 6-2.
Women's top seed Serena Williams won through to the quarter-finals despite falling ill after tasting her pet dog's dinner in her Rome hotel.
Her Yorkshire terrier Chip had been served a salmon and rice dish and Williams told followers on social media she had been tempted by the look of it, saying: "I ate a spoonful. Don't judge me."
The food tasted “a little bit like a house-cleaner thing,” Williams said in a video posted on Snapchat, adding that she had made an urgent trip to the bathroom a few hours later.
“I don’t think it’s consumable for humans. They should have wrote that,” added Williams, who showed no lingering ill effects in beating fellow American Christina McHale 7-6(7) 6-1.
Former world number one Federer still looks way short of his best after knee and back injuries and struggled to get to grips with the 22-year-old Thiem at the Foro Italico, a week after withdrawing from the Madrid Open.
Thiem, ranked 15th, recovered from an early break and streaked into a 6-2 lead in the tiebreak before taking the opening set with a backhand winner.
He broke the 34-year-old Federer's serve again midway through the second set as he claimed the 32nd victory of a breakthrough season that saw him rise to a career-high 13th.
Seventeen-times major champion Federer, who has missed 10 weeks of the season following knee surgery, shrugged off the defeat, saying his immediate priority was the French Open that begins on May 22.
"This is like, 'who cares about the results here?'. It matters what comes now in the next couple of months," Federer told a news conference.
"Now obviously time starts ticking more towards Paris. Clearly the way I'm playing right now is never going to be enough for a good run in Paris... I'm still confident I will be fine somehow."
Federer, a four-times finalist in Rome, missed last week's Madrid Open with a back problem but said there had been some encouraging signs this week.
"For the first time maybe I could play a match really playing freely, trying out a few things," he added.
World number one Djokovic was on the wrong end of a love set for only the ninth time in his career, picking up just eight points.
From there, though, he moved to a smooth victory over the 37th-ranked Belluci and a quarter-final meeting with Rafael Nadal, who defeated Australian Nick Kyrgios 6-7(3) 6-2 6-4.
World number three Andy Murray came back from a break down in the second set to defeat Frenchman Jeremy Chardy 6-0 6-4 and set up a meeting with David Goffin, who thrashed Tomas Berdych 6-0 6-0.
Women's third seed Garbine Muguruza of Spain reached the quarter-finals by beating Latvian Jelena Ostapenko 6-1 6-4.
(Writing by Martyn Herman and Clare Fallon; Editing by Pritha Sarkar)
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