Garin's attacking game paid off as he dispatched world number four Ruud in just under two hours.
Tenth-seeded Norrie, who reached the final in Buenos Aires and lifted the trophy in Rio de Janeiro last month, was down a set and 4-1 to Japanese qualifier Taro Daniel before rallying for a 6-7 (5/7), 7-5, 6-2 victory.
Garin, ranked 97th in the world, earned his first hard-court victory over a top-10 player and continued his impressive form in the California desert, where he hasn't dropped a set in qualifying or main draw matches at the ATP and WTA combined event.
"The way I played today I'm so happy," the 26-year-old said. "I was aggressive the whole match."
His 39 winners were too much for Ruud, who had hoped a quick victory over Diego Schwartzman in his second-round opener would mark a turnaround in a season where he hadn't advanced past the second round in three prior tournaments.
A two-time Grand Slam finalist in 2022, the 24-year-old Norwegian fell to Jenson Brooksby in the second round of the Australian Open.
He broke Garin's serve just once, to level the second set at 3-3, but the Chilean turned up the heat again in the tiebreaker.
"Aggressive, going to the net, I think that's the way that I like to play and the only way that I have to beat these kinds of players," Garin said. "I want to keep improving and keep enjoying this tournament."
Norrie, the 2021 Indian Wells champion, looked like he, too, might be headed for an unexpected exit after 103rd-ranked Daniel squeezed out the first-set tiebreaker and grabbed an early break on the way to a 4-1 lead in the second.
Daniel, with metronomic consistency from the baseline, looked poised to pull off a repeat of his 2018 Indian Wells victory over Norrie, when they met in the first round.
But Norrie, with the confidence of his recent results, was able to make the necessary adjustments and grabbed 15 points on 21 forays to the net in the second set then dominated in the third.
Ruud was the top men's seed in action Sunday, with fifth-seeded Daniil Medvedev -- coming off three ATP titles in as many weeks -- headlining the night session against Ilya Ivashka.
Australian Open champion Aryna Sabalenka of Belarus was due to cap the action on Stadium Court against Ukrainian qualifier Lesia Tsurenko.
In other early matches, 2022 women's runner Maria Sakkari shook off a slow start to beat Ukraine's Anehelina Kalinini 3-6, 6-2, 6-4.
Sakkari, the seventh seed, needed two and a half hours to seal her second straight come-from-behind victory.
She was down a break in both the second and third sets, but held on and will face either Veronika Kudermetova or Karolina Pliskova for a place in the quarter-finals.
Other women's matches on Sunday's slate included third-seeded American Jessica Pegula taking on Anastasia Potapova and fifth-seeded American Coco Gauff facing Czech Linda Noskova.
