Serbia’s Djokovic, who lost the Wimbledon final to Alcaraz, won 6-0 6-2 6-3 against France’s Alexandre Muller in Monday’s night session in New York.
Djokovic, 36, was not allowed to play in the United States last year because he was not vaccinated against Covid-19.
He has returned knowing another title will be a record-equalling 24th major.
Matching Australian Margaret Court’s all-time tally is Djokovic’s target and he made a statement start by breezing past 84th-ranked Muller.
“We started quite late but nevertheless I was excited to go out on the court – it has been a couple of years. In the second and third sets I dropped the level of my serve and I had to work for the points a bit more,” said Djokovic, who will spend a record-extending 390th week at the top of the ATP rankings.
Djokovic, who won the Australian Open and French Open earlier this year, is aiming for a fourth title at Flushing Meadows and will play Spain’s Bernabe Zapata Miralles in the second round. He will return to world number one after the tournament.
Alcaraz and Djokovic are the joint favourites for the men’s singles title having developed a fascinating rivalry over the past 12 months.
The match-up has ignited over the past few months following three compelling contests in the French Open semi-finals, the Wimbledon showpiece and this month’s Cincinnati final.
Alcaraz and Djokovic cannot meet at the US Open until the final as a result of being the top two seeds.