Djokovic won 6-1, 6-7 (5-7), 6-2, 6-1 as Rune, 18, was hampered by cramps from the third set of his Grand Slam debut.
German fourth seed Alexander Zverev started his bid with a straight-set victory against American Sam Querrey.
The 24-year-old – who reached the final in 2020 – wrapped up his routine 6-4, 7-5, 6-2 win in one hour and 40 minutes.
Zverev is looking to add a first Grand Slam title to the Olympic gold he won in Tokyo at the start of August.
Djokovic’s bid to become the first man to win a ‘Golden Slam’ – all four Grand Slams and the Olympic title in the same year – was ended by Zverev in the Tokyo 2020 semi-finals. But the world number one is still the heavy favourite to win at Flushing Meadows and become only the sixth player to complete a clean sweep of the Australian Open, French Open, Wimbledon and US Open. (BBC)