A dark horse at the year's first Grand Slam after his Paris Masters final victory over Novak Djokovic in November, Rune moved into the last 16 with a 6-4 6-2 7-6(5) win to book a fourth-round spot against Andrey Rublev.
With home favourite Nick Kyrgios absent due to a knee problem, holder Rafael Nadal and second seed Casper Ruud limping out in the second round, and nine-times champion Novak Djokovic hampered by a hamstring issue, organisers would have been holding their breath when another seed took a tumble.
Looking dazed after turning his ankle and landing awkwardly, Rune needed lengthy treatment on court, but the Dane is made of sterner stuff.
"It wasn't too bad," he told reporters. "I was able to play full. The third set, of course, I was a bit worried by what happened, but I saw the slow-motion after the match. Nothing dangerous.
"I relaxed a lot more because I was a bit afraid, but honestly, I played very good after it happened. So I'm feeling hopeful."
Rune had reached the third round with wins over Filip Krajinovic and Maxime Cressy and is yet to lose a set, but Russian fifth seed Rublev presents a tough obstacle.
Rublev said the Dane would not go into the match as the underdog after a breakthrough year in which he took out a number of top-10 players, most notably during his Paris title run.
Rune agreed.
"Honestly, I never feel like an underdog," Rune said. "I have high ambitions every time I play. Even we played each other last year in Paris. I was still believing that I could beat him, and I did beat him.
"He's obviously playing well here. He won in straight sets today as well. And he's really solid. Like, forehand, backhand he can do anything on the tennis court. He has no weakness.
"I'll be expecting a tough match, but really looking forward to it."