The big-serving Frenchman's aggressive game forced world number one Alcaraz to call on every shot in his arsenal to turn a tense contest his way.
Alcaraz will face 13th-seeded Norwegian Casper Ruud for a place in the quarter-finals.
"Really happy that I was able to turn things around," said Alcaraz.
The Spaniard nearly managed to steal the first-set tiebreaker, in which he had trailed 2-5, but a wayward forehand on set point gave Rinderknech his chance to pocket the set with a pair of winners.
Rinderknech then produced the first service break of the match to put the top seed in a hole in the opening game of the second set.
Alcaraz responded with an immediate break back, ramping up the pressure on Rinderknech's serve on the way to a decisive break in the sixth game.
When Alcaraz broke again to open the third set he was on his way, rolling to his sixth win over 26th-ranked Rinderknech in six career meetings.
"In the beginning I was a little bit mad with myself because I just had a lot of chances in the first set, a few break points, set point, and I couldn't make it," Alcaraz said.
"I got in trouble to be honest. But I'm just really happy with the way that I just dealt with everything that was happening, accepted it, kept going, stayed strong mentally."
Rinderknech had never won a match in Indian Wells in three prior appearances and he leaves this edition without an on-court victory after enjoying a first-round bye and advancing to the third round by walkover.
Alcaraz, 22, became the youngest man to complete a career Grand Slam with his victory at the Australian Open.
He followed that up with the Qatar Open title and with two wins under his belt in Indian Wells is now 14-0 on the season.
He next faces Ruud, who beat Valentin Vacherot 3-6, 6-3, 6-4.