Serena Williams is "looking better every day" and only she could handle the US Open hoopla that surrounds her, according to world number one Iga Swiatek.

In a febrile atmosphere, Williams made a winning start to her final Flushing Meadows campaign on Monday night, getting the better of Montenegrin Danka Kovinic.

A 6-3 6-3 first-round win over the world number 80 prompted Williams to break out in a jig of delight, and now world number two Anett Kontaveit awaits the great American in round two.

Williams will returns to Arthur Ashe Stadium on Wednesday evening, when she is certain of another big reception and overwhelming support.

Kovinic's best winners barely received a smattering of applause, while her mistakes were often cheered by some in the crowd.

The crowd's partisanship could again come into play when 23-time grand slam singles champion Williams, who is planning on retiring after this tournament, tackles a tough task against Kontaveit.

Swiatek watched the opening games on Monday and remarked that she had "got tight just by watching", feeling all the strain Williams was under.

"So I realised that I need to stop," said a smiling Swiatek. "I can't imagine what she must feel, having this kind of atmosphere around her and just finishing.

"I know she's doing a tournament, but I feel like yesterday it was like people did so much work to show appreciation to what she's done. It's pretty amazing. I have never seen something like it.

"For sure it was like the most popular first round of a slam ever. I'm pretty happy that she can experience something like that. I also feel that not every player would handle that kind of fuss around your first match of the tournament.

"She's handling it pretty well, as usual. So that's just confirmation of how great she is."

Williams came onto the court in a glittery tennis dress and cape, along with jewel-encrusted tennis shoes, with Swiatek describing the outfit as "pretty cool".

With two grand slam titles to her name, the 21-year-old Swiatek is among the favourites to be champion in New York.

When she watched the fanfare and ceremony for Williams, along with the first couple of games, it brought it home to Swiatek just why many regard the 40-year-old as the GOAT – greatest of all time.

"It was kind of too much for me even watching that. I realise how she must feel, but she is kind of used to it more than us," Swiatek said.

"We never had such attention around us, because she's just a GOAT. She has it probably for most of her career, so I guess she's used to it. But for me, just watching that, I was pretty shocked."

Swiatek refused to predict how the Kontaveit match would go. The Estonian is ranked second largely because of her strong run of form at the back end of last season, with recent results not so impressive or consistent.

"I feel like against Serena – I have never experienced that, so I just assume – it's all about your mindset and mentality," said French Open champion Swiatek, who beat Italian Jasmine Paolini 6-3 6-0 on Tuesday.

"I know Anett can play great tennis. It's not a mistake that she's second in the world right now. But on the other hand, I feel like Serena is looking better and better every day."