The sixth seed got off to the perfect start, winning his opening round-robin match in the Pete Sampras Group as he was able to keep his nerves in check to beat the Austrian 6-3 5-7 7-5.
The 26-year-old looked to have blown his chance to win when failing to serve out the match, but Thiem then produced two double faults in a row to hand the momentum back to Dimitrov.
The Bulgarian continued to make sloppy errors but did enough to overcome his rival and he can now look forward to more matches this week.
"I'm not going to lie - I was pretty nervous, my first match out here," he told reporters. "I'm just very grateful to win that match, especially in that manner. It's never easy to come out here and play for the first time."