Thiem easily advanced in just under an hour. The 23-year-old Austrian's biggest challenge occurred in the ninth game of the second set when he was denied five match points before finishing the match.
"He's a very tricky opponent, it's really different to play against him. You cannot really prepare, but I had a very good start," Thiem said. "In general, I returned very well, probably better than I expected myself. It's very important the first two shots against him, return and then the second, because he's covering the net really, really well."
A win in his next match would mark the first time Thiem advanced to the quarterfinals at Indian Wells. To reach that stage, he would need to get past 10th-seeded Gael Monfils of France.
Zverev stunned Andy Murray before falling to Roger Federer in the Australian Open quarterfinals. Since his surprising run, Zverev had lost four straight matches before winning his opening-round match in Indian Wells against Joao Sousa of Portugal.
Monfils had no difficulty dispatching 20th-seeded American John Isner 6-2, 6-4 in 68 minutes. The Frenchman never faced a break point in the match, and he broke serve on three of his eight opportunities.
Spain's Pablo Carreno Busta earned a free pass into the fourth round as compatriot Roberto Bautista Agut withdrew ahead of their match due to an abdominal injury.
Carreno Busta will face Dusan Lajovic of Serbia. Lajovic knocked off Canadian Vasek Pospisil 6-7 (4), 6-3, 7-5 Monday night.
Pospisil advanced to the third round by stunning Murray in the second round on Saturday.
Belgium's David Goffin, the 11th seed, shrugged off 22nd-seeded Spaniard Albert Ramos-Vinolas 7-6 (3), 6-4. Uruguay's Pablo Cuevas, seeded 27th, had little trouble with Italy's Fabio Fognini, posting a 6-1, 6-4 decision.
Yoshihito Nishioka of Japan, a "lucky loser" from qualifying who landed a spot in the main draw, continued his run by upsetting 13th-seeded Tomas Berdych of the Czech Republic 1-6, 7-5 (5), 6-4.