Norrie was runner-up at the San Diego Open earlier this month and reached the last eight of a Masters 1000 event for the first time by battling past Tommy Paul in three sets.
The 26-year-old, though, wasted little time in taking the early initiative against 11th seed Schwartzman on Thursday.
Norrie raced into a 4-0 lead against the world number 15 before another break and hold saw him take the opening set to love in just 31 minutes.
Schwartzman eventually got on the board in the opening game of the second set.
Norrie, though, gained an early break when Schwartzman sent a return long only to then fail to hold himself at 2-2.
Schwartzman was then broken again in the fifth and seventh games, swinging the momentum back towards his British rival, who closed in on victory.
Norrie served out his first match point to love when another Schwartzman return went long, completing a comprehensive 6-0 6-2 victory in one hour and 13 minutes.
Speaking after the win, Norrie said: “Definitely my biggest match of my career probably leading in. Once the match had begun, I was feeling very relaxed, and I was very in the present. I wasn’t really thinking too much and just going out and playing just like another match.
“I thought it was always going to be tough against Diego, and happy to play the way I did. I think that was the most impressive thing for me – 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 today.”
As a result of his run to the semi-finals, Norrie is set to break into the world’s top 20 – and so replace Dan Evans as the British number one.
Norrie was pleased to take over top spot but insisted he will continue to strive for further success, saying: “It was never really a goal of mine, but it’s definitely a great bonus to be British number one.
“I want to keep pushing. I think I’ve got a lot of things to improve on, but I think it’s one of those things you’ve got to enjoy.”