The eighth-ranked Frenchman, who won the event in 2011, missed a match point at 5-4 in the final set but converted his second chance in the tiebreaker after 2 hours, 15 minutes as the Austrian hit a forehand long.
"I was really poor from the baseline but I served well," said Tsonga, who had 16 aces. "I have played many matches like this so I have the experience. That made the difference. I am not in a good shape, I have to fight to win matches."
The 20-year-old Thiem, who played in only his second quarterfinal, said the match was "unbelievable."
"A third-set tiebreaker means the match could have gone either way," Thiem said. "I am proud I could match a top-10 player."
Haas reached his seventh semifinal of the season by defeating fifth-seeded Radek Stepanek of the Czech Republic 7-6 (10), 6-3. The victory kept the 12th-ranked German in the running to qualify for next month's ATP World Tour Finals in London.
"I live from match to match," said Haas, who has never before played in the season-ending event.
Haas failed to serve out the opening set at 6-5 but won the tiebreaker after saving four set points for Stepanek. He converted his only break point in the second set.
"It was mentally and physically tough," the second-seeded Haas said. "The tiebreaker was really great from both of us. I felt it was important to win it. If I'd lost it, I would have been very frustrated."
Haase, who won both his career titles in Austria, rallied to defeat third-seeded Fabio Fognini 6-7 (4), 6-1, 6-1 to advance to his fourth semifinal of the season.
Haase didn't face a break point in the entire match. Fognini had 32 baseline errors and ended the match with his second double-fault.
"I had my chances in the first set as well but lost it as he played an incredible passing shot," Haase said. "When I was 3-1 up in the decider and won five straight points from 40-0 down, I knew I wasn't going to lose this match."
Earlier, Rosol beat Ruben Bemelmans 6-3, 1-6, 6-3. Rosol broke the 176th-ranked qualifier once each in the first and final sets to reach his second career semifinal, and first since winning his maiden title in Bucharest in April.