In typically charismatic fashion, he produced his full repertoire of silky shot-making to win 6-3 7-6(3) 6-4.
Rune became only the fourth Dane to reach the men's third round at Wimbledon and the first for nearly 20 years.
"He made it very difficult and didn't give me anything for free, but I think I managed to raise my level and close to the end I got unbelievable support from the crowd," Rune, who twice had to recover service breaks in the opening set, said on court.
"I love getting the fans involved, it was unbelievable, really cool and I hope we can bring it all the way."
Rune's game style and casual demeanour have made him a hit with young fans who chanted "Ruuuuune" at regular intervals, and he responded with some exquisite winners and won many points with cunningly disguised drop shots.
"Grass is not an easy adjustment but it's a beautiful surface and to play on is beautiful - if you're playing well," he said.
