Eighth seed Jack Sock booked a place in the semi-finals of the Citi Open on Friday.
Sock, rated 19th in the latest ATP Tour rankings, captured a tight first set, 7-4, at the Rock Creek Park Tennis Center in Washington, D.C., before toppling Raonic 6-4 in the second set to secure a victory.
In other women's matches, top-seeded Simona Halep came back from losing the first set to beat Mariana Duque-Marino 3-6, 6-4, 6-2, and Andrea Petkovic beat Eugenie Bouchard 6-2, 4-6, 6-0.
After beating sixth-seeded French star Gael Monfils and Argentine Guido Pella on his way to the quarter-finals - his career-best run at an ATP 500 event - Bhambri would have looked to continue the good momentum but he got broken twice in the first set as he went a set down 35 minutes into the match.
The quarterfinal matchups are Anderson vs. Bhambri, Raonic vs. Despite some shaky serving from both players, the South African 31-year-old giant served up 19 aces despite some double faults, and was more comfortable than his opponent on his native surface.
"I've practised with him before, but that was a few years ago".
Thiem staved off defeat the first time after an 11-stroke rally culminated with a whipped forehand victor into the corner that Anderson could only watch helplessly as it whistled past.
Anderson would use that height and serve of his to break back, making it 5-5. He won the Citi Open in 2014 when he defeated Vasek Pospisil for the championship.
Anderson didn't want to waste a challenge over the call though and Thiem would concede the point.
"Hopefully I can win a few more matches here", Bhambri said. At 5-6, Thiem had a match point, but would not be able to convert.