NEW YORK—Venus Williams took to Arthur Ashe Stadium on Thursday afternoon looking for her 70th career U.S. Open victory. The world No. 6 had little trouble reaching that milestone, defeating Germany’s Julia Goerges, 6-2, 6-3, and advancing to the third round.

Williams is playing in her record-setting 72nd Grand Slam, and to say that she’s comfortable in Arthur Ashe Stadium is a vast understatement. While the rain disrupted play on all other courts, these two got to close out the day session under the new roof, in front of a happily dry crowd.

In the first set, the seven-time Grand Slam champion drew first blood, edging ahead of her 27-year-old opponent 4-2. The world No. 64 struggled to hold her serve throughout the match, winning just 53 percent of her first-serve points in the first set.

Goerges also hit more than twice as many unforced errors as Williams. One of the 36-year-old’s biggest goals was to keep her unforced errors down. After her three-set win on Tuesday, she noted her tendency to spray shots against opponent Kateryna Kozlova.

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“I definitely had a lot more errors than I wanted,” Williams said on Tuesday. “If I could cut those in half, it's definitely a different story. The good part is I'm playing the game I want to play. I'm playing aggressively and moving forward. It's just about making a few less errors, and it's a completely different story.”

Williams had never faced Kozlova before.

Things went differently on Thursday. Williams had played Goerges just once prior, in Wuhan last year, and prevailed in straight sets in that match. She came out on Thursday showing no mercy, breaking serve three times to take the set 6-2.

In the second set, Goerges—returning from a mid-summer hip injury—still struggled to hold serve. Williams, on the other hand, looked impenetrable, making just one unforced error to go up 4-1.

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Arthur Ashe Stadium is a fitting arena for a player like Williams, who stays so calm and so composed regardless of the atmosphere around her. She steadily went about her business in front of a New York crowd that openly adores her.

“There’s nothing like playing in New York,” Williams said.

Williams’ return game was relentless against Goerges, but two aces helped the German narrow the gap to 2-4. The American was up a set and a break against Kozlova before losing her rhythm, and the last thing she wanted on Thursday was to go to a third set. On this day, not only did she limit her errors, but she closed exceptionally well. It was as if she spent the time in between matches improving on everything that went wrong against Kozlova on Tuesday.

“I just want to keep playing aggressive and make less errors,” she said. “That’s definitely a success in this match. As long as I’m walking to the net, and I shake hands and I’ve won the match, that’s all I need.”

Instead of dropping her level down the stretch, Williams saved two break points to go ahead 5-2. Goerges started to go for broke, even pulling off a second-serve ace to cut it to 5-3. In the ensuing game, her formidable serve bailed her out. Williams saved two break points and sealed a spot in the third round.