History-chasing Djokovic, who has won eight of the past 12 Grand Slam events, will next face 121st-ranked Dutchman Tallon Griekspoor, a rival the top seed admitted he didn't know much about.
"I know this court well," Djokovic said. "Hopefully that's going to help."
Djokovic would complete the first men's singles calendar-year Slam since Rod Laver in 1969 by winning the crown on the New York hardcourts.
He would also break the deadlock for all-time men's Slam titles at 20 which he shares with Roger Federer and Rafael Nadal, both absent due to injuries, as is defending champion Dominic Thiem.
Rune, ranked 145th, began cramping in the third set and struggled to finish.
"From the third or fourth game in the third set, he couldn't move much," Djokovic said. "You could see he struggled a lot. I feel for him."
Djokovic won the first set in 26 minutes without facing a break point, fell behind 3-0 to start the second before battling into the tie-breaker, where Rune grabbed leads of 4-0 and 6-3. After two Djokovic service winners, he sent a backhand service return long to drop the set.
In the third set, Djokovic broke to lead 3-1 and Rune began getting treatments between games, grimacing at times and struggling to run for shots to the end.
Zverev stretches streak
Earlier Tuesday, Tokyo Olympic champion Alexander Zverev stretched his win streak to 12 matches and warned he has his eyes on denying Djokovic the Slam.
Fourth seed Zverev, who beat Djokovic in an Olympic semi-final, dispatched American Sam Querrey 6-4, 7-5, 6-2.
"I hope in two weeks' time I'll be on an 18-match winning streak," said Zverev, who would claim his first Grand Slam title if he creates such a run.
The 24-year-old German, last year's US Open runner-up, hit 18 aces and 40 winners while never facing a break point in advancing after an hour and 40 minutes.
"Novak is chasing history," Zverev said. "But the young guys are going to try and get in the way of that. I'm going to try to give him a challenge as well."
Zverev, who next faces Spain's Albert Ramos-Vinolas, could face Djokovic in the semi-finals.
"I hope I can keep the level up," Zverev said. "To beat Novak here is going to be an extremely difficult task."
Qualifier pulls shocker
French-born US qualifier Maxime Cressy fired 44 aces and saved four match points to upset Spanish ninth seed Pablo Carreno Busta 5-7, 4-6, 6-1, 6-4, 7-6 (9/7).
After beating Djokovic for a Tokyo Olympic bronze medal, the Spaniard's US Open ended slamming down his racquet in frustration.
"I'm really happy things are starting to pay off," Cressy said.
World number one Ashleigh Barty, coming off wins at Wimbledon and Cincinnati, began her chase for back-to-back Slam titles by defeating Russian Vera Zvonareva 6-1, 7-6 (9/7).
"When my back was against the wall late in that buster, I came up with some really good stuff," Barty said.
The 25-year-old Australian next faces Denmark's Clara Tauson as she goes for her sixth title of the year.
Japan's Kei Nishikori, the 2014 US Open runner-up, eliminated Italy's 113th-ranked Salvatore Caruso 6-1, 6-1, 5-7, 6-3. He next plays American Mackenzie McDonald with Djokovic a possible third-round opponent.
Italian sixth seed and Wimbledon runner-up Matteo Berrettini, a 2019 US Open semi-finalist, defeated France's Jeremy Chardy 7-6 (7/5), 7-6 (9/7), 6-3.
Canadian sixth seed Bianca Andreescu, the 2019 US Open winner who skipped last year's event, struggled but outlasted Swiss Viktorija Golubic 7-5, 4-6, 7-5.