The British number one is continuing to live up to his top-10 ranking and kept 19-year-old Rune at arm’s length during a 7-5 6-4 6-1 victory at Flushing Meadows.
Rune, already the highest-ranked Danish man in history and a French Open quarter-finalist this year, has a reputation as a spiky character and during the second set he protested to the umpire about Norrie catching his ball toss before serving.
Players are allowed 25 seconds between points but the clock stops when the server throws up the ball, meaning catching the toss buys extra time to reset.
Rune made it clear to the umpire he thought Norrie was deliberately exploiting a loophole in the rules but his protests fell on deaf ears as the seventh seed maintained his record of not having dropped a set so far this fortnight.
Norrie, whose run to the Wimbledon semi-finals was the first time he had gone beyond the third round at a Slam, had admitted to nerves during his second-round win over Joao Sousa and both players looked edgy in a scrappy first set.
Norrie opened up a 5-3 lead but hit a wild double fault and then shanked a smash to drop serve, only for Rune to play a similarly poor game and give him a second chance.
The British number one looked to have taken control of the second set at 4-1 but Rune was able to apply some late pressure again, forcing Norrie to save two break points at 5-4.
The Dane complained to the umpire about Norrie’s ball toss during that game before continuing his protests at the end of the set.
The pair had met twice before, with both matches going to a deciding set, but here Rune could offer no real resistance in the third as Norrie eased to victory.