Second seed Tsitsipas told reporters on Wednesday that he did not expect to make a deep run at the tournament due to his ailing shoulder and it was unclear how much he had left in the tank after he was blanked in the first set tie-breaker.
But the Greek battled back in the second and pumped up the crowd ahead of the third set tie-break.
But overly aggressive play in the tie-breaker, including a forehand on match point that missed the line by the slimmest of margins, was his undoing.
A beaming Thompson fired a ball high into the air in celebration after he notched his first win over Tsitsipas in three career meetings and second over a top 10 player.
"Tricky match point," Thompson told reporters.
"The ball was only just out and it was a relief that it went out. It was a great feeling. You could see it on my face at the end of the match. I was overcome with joy."
NORRIE ADVANCES
Earlier, Briton Cameron Norrie cruised past Taiwan's Tung-lin Wu 6-2 6-4.
Norrie, the 2021 champion, broke Wu four times en route to the win and will next face either Italy's Matteo Berrettini or Japanese qualifier Taro Daniel, who play later on Friday.
"Lots of good memories at Indian Wells," Norrie said. "It's nice to be through to the third round."
The 10th seed is off to a terrific start this year having beaten world number two Carlos Alcaraz in the Rio Open final last month to claim his fifth singles title.
Norrie's win comes a day after unseeded Britons Andy Murray and Emma Raducanu reached the second round at the ATP and WTA 1000 event in the Southern California desert with gutsy wins.
Norwegian Casper Ruud's serve-forehand one-two punch proved too much for Diego Schwartzman as the third seed advanced 6-2 6-3 over a player who has given him trouble in the past.
"Everything sort of went my way," said Ruud, who now has four wins and five losses against the Argentine.
"Early on there were a couple of line shots and net cords that went my way, so I kind of felt that maybe today is my day."
SABALENKA SLAYS
On the women's side, second seed Aryna Sabalenka needed just over an hour to dispatch Russia's Evgeniya Rodina 6-2 6-0 as the Australian Open champion improved to 14-1 on the year.
"Happy to win this match in two sets without struggling too much," Sabalenka told reporters.
The Belarusian next faces Ukrainian Lesia Tsurenko. Players from Belarus and Russia are once again playing under neutral flags at the tournament due to Russia's invasion of Ukraine.
"It's going to be an interesting match," Sabalenka said.
"She's a great player and it's going to be a great battle."
Swiss Jil Teichmann upset her doubles partner and countrywoman Belinda Bencic 3-6 6-3 6-3 on the Tokyo Olympic champion's 26th birthday to reach the third round for the first time.
Third seeded American Jessica Pegula roared back to beat Italy's Camila Giorgi 3-6 6-1 6-2 under the lights on center court to set up a third round showdown with Russia's Anastasia Potapova.
Elsewhere, Ukrainian Lesia Tsurenko upset Croatian Donna Vekic 2-6 6-2 6-2, Latvian Jelena Ostapenko defeated Aliaksandra Sasnovich 7-5 3-6 6-2 and Czech Barbora Krejcikova breezed past Ukrainian Dayana Yastremska 6-1 6-2.
