Norrie needed just 73 minutes to dismantle Schwartzman, who finally held serve to open the second set, drawing cheers from the small crowd at Indian Wells Tennis Garden, where attendance has been capped at well below capacity because of the coronavirus pandemic.
"For my biggest match of my career to play like I did and execute like I did was the biggest thing that I'm most happy with," Norrie said.
No. 12 seed Ons Jabeur advanced to the semifinals with a 7-5, 6-3 win over Anett Kontaveit that will move the 27-year-old Tunisian into the world's top 10 for the first time next week. She's projected to be No. 8, but could rise as high as No. 5 by winning the tournament.
"Feels amazing," Jabeur said. "We've been working hard for years. The end of last season I said I wanted to be in top 10. It means a lot, but we're not going to stop here and go further hopefully."
Jabeur, runner-up in Chicago recently, won her WTA Tour-leading 48th match of the year.
"I'm proving more things and now that I'm a top-10 player, I need to be more solid," she said.
Jabeur beat Kontaveit for the third time in four career meetings. Since losing in three sets to Jabeur at Cincinnati in August, Kontaveit had won 16 of 17 matches, including 14 in straight sets, while claiming two titles.
Badosa beat 15th-seeded Coco Gauff in three sets and No. 3 Barbora Krejcikova in straight sets on her way to playing Jabeur in the semifinals on Friday. The other semi pits Victoria Azarenka against Jelena Ostapenko.
"Why not get the title here?" Jabeur said.
Schwartzman earned his only break of the match to tie it 2-all in the second. Norrie ran off the final four games to move into a semifinal against 23rd-seeded Grigor Dimitrov, who beat No. 8 Hubert Hurkacz 3-6, 6-4, 7-6 (2).
Schwartzman's serve deserted him under cloudless skies. He won 13 of 27 points on his first serve, and just 3 of 19 on his second serve to go with four double faults.
Hurkacz dropped a set for the first time in the tournament, losing on a let cord. Dimitrov held up both hands in a gesture of apology.
Hurkacz trailed 5-2 in the third before winning three straight games to tie at 5-all. He held at 6-all to force the tiebreak. But he appeared exhausted at that point, committing unforced errors on the final three points to lose.
"I was clearly not at my strongest, but I felt throughout the whole time I was able to do something with the ball even when I was tired," Dimitrov said. "That really made a big difference."
Norrie won his career-best 45th match of the year against 20 losses in reaching his first Masters 1000 semi. He'll become the No. 1 player in Britain in next week's rankings.
"It was never really a goal of mine, but it's definitely a great bonus to be British No. 1," Norrie said. "I want to keep pushing. I've got a lot of things to improve on."
In quarterfinal matchups Friday, No. 2 Stefanos Tsitsipas plays Nikoloz Basilashvili and No. 3 Alexander Zverev faces Taylor Fritz. Tsitsipas and Zverev are the highest seeds remaining in the men's draw after Dmitrov beat Daniil Medvedev, the top seed and US Open champion.