Before the match, left-hander Norrie had said he was “concerned” by a problem with his left wrist which forced him to pull out of an ATP event in Auckland five days ago.
Norrie, who is Britain’s highest-ranked singles player, is safely through and faces Italian qualifier Giulio Zeppieri next.
After a poor end to last year where he admitted feeling burned out, Norrie is looking to bounce back in 2024 and go beyond the third round in Melbourne for the first time.
After scrapping to save a break point in his first service game, Norrie quickly asserted his authority as he took advantage of Varillas’s poor service game.
Norrie applied heavy pressure as the Peruvian barely landed a first serve, eventually taking a sixth break point for a 3-2 lead which was enough to secure the opening set.
With Varillas’ first-serve percentage down below 45%, Norrie moved a double break up at the start of the second set, allowing for him to drop serve before seeing it out at the second opportunity.
Varillas produced a double fault to hand over another break for 2-1 in the third and Norrie continued to move through the gears nicely.
He took his opponent’s serve for the fifth time before ending Varillas’ late resistance by taking his third match point in what was a pleasing opening performance.
“I think I did the basics well. I served well in the big moments and I think I won a lot of points on my first serve when I did make them,” Norrie said.
“I didn’t think I served that well, but I didn’t lose many points on my first serve, which was good to see.”