NEW YORK — In Coco Gauff’s first match since hiring someone to assist with her problematic serve, she faced a challenging beginning at the U.S. Open on Tuesday night. She committed her first double-fault right in the opening game, accumulating 10 in total. Furthermore, she lost her serve in that game and was broken six times throughout the match.

Ultimately, the numbers that mattered most were displayed on the Arthur Ashe Stadium scoreboard. There, they reflected that Gauff, the No. 3 seed, endured to secure a 6-4, 6-7 (2), 7-5 win over Ajla Tomljanovic, advancing her to the second round at Flushing Meadows.

“It wasn’t the best,” Gauff said, “but I’m happy to get through.”

The match was anything but simple. Gauff led by a break twice in the second set but couldn’t close it out. Leading 5-3 in the third set and serving for the match at 5-4, she double-faulted twice consecutively and missed two forehands, evening the score to 5-all.

“Staying close also puts her under pressure to serve it out,” Tomljanovic said. “In the first round, even if you’re Coco, it’s never easy.”

This misstep could have been detrimental for Gauff. However, she quickly regained her composure, broke serve immediately after, and succeeded in serving out the match on her second attempt, nearly three hours after the battle began.

“I had so many chances. … I was just like, ‘Eventually, one of these is going to go my way,’” she said.

Before this tournament, Gauff, who claimed her first of two Grand Slam titles at the 2023 U.S. Open during her teenage years, added Gavin MacMillan to her coaching crew. MacMillan specializes in biomechanics and previously helped current World No. 1 Aryna Sabalenka improve her serve. During the match, he was seated in the front row of Gauff’s guest box, right in front of her mother.

After beating Tomljanovic, Gauff called her practices with MacMillan “really tough” and “mentally exhausting.”

“I was spending lots of time on the court, literally serving to the point where my shoulder hurt,” Gauff explained. “I feel like it’s headed in the right direction, and my goal is to avoid reverting to old habits during pressure points, which I think I managed well today, particularly in the third set.”

The problem for Gauff, in a nutshell, has been a propensity to accumulate double-faults. Her 320 entering the U.S. Open were the most on the women’s tour this season — and more than 100 more than anyone else. That included 23 in one match earlier this month, then 14 the next time out.

On Tuesday, as she dealt with the work-in-progress of a tweaked service motion, Gauff began with much slower offerings than she’s capable of striking. As the match progressed, and the tension rose, the 21-year-old from Florida reverted to her customary pace, going from averaging just 88 mph on first serves in the first set, to 97 mph in the second and 101 in the third, when Ashe’s retractable roof was closed. She cranked one in at 117 mph and even produced one second-serve ace.

What won this one against the 79th-ranked Tomljanovic, an Australian best known for defeating Serena Williams at the 2022 U.S. Open in the 23-time major champion’s final match of her career, was Gauff’s exemplary court coverage and terrific backhand. Appropriately, a down-the-line backhand converted match point, and Gauff waved her arms overhead to rile up the crowd.

Even as Tomljanovic swung away on her big forehand, it was Gauff who got the best of their lengthy exchanges from the baseline.

She also fared well when she pressed forward, winning 12 of the 15 points she ended at the net, including one with a leaping, over-the-shoulder, back-to-the-net volley winner in the third set

This was Gauff’s first match at a Slam since stumbling to a first-round exit at Wimbledon in July, a setback that followed her championship at the French Open in June.

As Gauff moves forward at Flushing Meadows, there is a chance she can overtake Sabalenka and No. 2 Iga Swiatek in the rankings and rise to No. 1 for the first time.