The season's final Grand Slam began on Monday with a change at the top after Novak Djokovic demolished Frenchman Alexandre Muller to ease into the second round - a result that meant the Serb will replace Alcaraz as world number one when the rankings are updated on September 11.
But Alcaraz will have the number one attached to his name for the duration of the Flushing Meadows fortnight and advanced to the second round after Koepfer twisted his ankle minutes into the match and later retired with the Spaniard up 6-2, 3-2.
Alcaraz, bidding to become the first man to retain his US Open crown since Roger Federer won five straight from 2004 to 2008, appeared unsteady at times on the North American hardcourts ahead of the season's final Grand Slam but kept his composure on Tuesday with few errors.
"I'm not thinking about defending the title. I was not thinking about I was the champion last year," he told reporters.
"I just focus on play my best level, to recover the level that I played last year, try to do same things that I did last year. That's the only thing that I'm thinking right now."
With Alcaraz and Djokovic hogging the spotlight, Daniil Medvedev had been the forgotten man but the third-seeded Russian remains a threat to claim his second US Open title in three years.
He got everyone's attention when he dispatched Hungarian Attila Balazs 6-1, 6-1, 6-0 in a speedy affair to kick off the action on Arthur Ashe Stadium, firing off 41 winners to set up a second round meeting with Australian Christopher O'Connell.
Andy Murray, the 2012 US Open champion, recorded a landmark 200th Grand Slam match win as he turned back the clock to beat young Frenchman Corentin Moutet 6-2, 7-5, 6-3.
The Scotsman said he was playing some of the best tennis on a consistent level since 2017. He faces Bulgarian Grigor Dimitrov in the second round.
Italian sixth seed Jannik Sinner handily beat German Yannick Hanfmann 6-3, 6-1, 6-1, and will play compatriot Lorenzo Sonego next.
Second seed Aryna Sabalenka sent off Belgian Maryna Zanevska 6-3, 6-2, closing out the action on Louis Armstrong Stadium with 27 winners to set up a meeting with Britain's Jodie Burrage.
"I think she played an unbelievable match - she fought until the end," the Australian Open champion said in on-court remarks. "I think I'm most pleased with my focus... Hopefully I'll do better with every game."
American Jessica Pegula took her place in the US charge into the second round as she fired off eight aces to swat aside Italy's Camila Giorgi 6-2, 6-2.
The third seed has taken a backseat to compatriot Coco Gauff in the headlines, as local fans hope to see an American hoist the trophy for the first time since 2017, but said she has no problem ceding the spotlight.
"Coco can take it, that pressure, not me," said Pegula, who will play Romanian Patricia Maria Tig next. "It's nice that there is a lot of us that are doing well and that have chances to go deep here."
Seven-time Grand Slam winner Venus Williams came up short of her goal to make it past the opening round for the first time since 2019, losing 6-1, 6-1, to Belgian qualifier Greet Minnen in her 100th U.S. Open match.
"First I have to give credit to my opponent," Williams told reporters.
"There wasn't a shot she couldn't make. Even when I hit really amazing shots, she just hit a winner or a dropshot, so I don't think I played badly. I think it was just one of those days where it's just unlucky."
Asked when she might hang up her racket, Williams was non-committal.
"I wouldn't tell you," she said with a smile.
"I don't know. I don't know why you're asking."
