Simona Halep battled to victory over Elena Rybakina in the third round of the US Open but last year’s finalist Victoria Azarenka was beaten by Garbine Muguruza.

To have two such high-level encounters so early in the tournament highlights the depth of women’s tennis, and both lived up to their billing.

Halep’s season has been blighted by injuries, and the Romanian is playing with heavy strapping on her right thigh, but she saved four set points in a remarkable first-set tie-break and finished strongly for a 7-6 (11) 4-6 6-3 victory.

Halep, who was down an early break in all three sets, said: “Before the match, I was super stressed and I told (coach) Darren (Cahill) that my forehand is lost. So I cannot feel it anymore.

“I had emotions. I was nervous. And I was crazy a little bit on court, as well, the frustrations got me a little bit. But I fought, and this victory gives me a lot of confidence that my game is coming back and also the fighting spirit is there.

“So I feel safer on court when I step now. I know that every match is a battle but I’m there and, if I’m healthy, I’m confident that I can play my game.”

Last year’s tournament marked the return of Azarenka as a grand slam force, with the former world number one defeating Serena Williams before losing to Naomi Osaka in the final.

But ninth seed Muguruza proved too strong here, the Spaniard prevailing 6-4 3-6 6-2 to set up another blockbuster clash against French Open champion Barbora Krejcikova.

Halep said with a smile: “Actually last night at dinner we were looking at the schedule, and I was, like, ‘Every match is a semi-final’.

“The level is super high, and now you cannot expect some players to win. I gave up thinking who is going to be in the semis and finals, because I like to discuss these things. The matches are great, in my opinion, for everybody.”

Krejcikova, whose rise has been so rapid that this is her main draw debut in New York, was a comfortable 6-4 6-2 winner over Kamilla Rakhimova.

A day after cheering on husband Gael Monfils to a second-round win, fifth seed Elina Svitolina made it through to round four with a 6-4 6-2 victory over Daria Kasatkina.

Another big-name encounter saw two former champions meet, with Angelique Kerber continuing her resurgent summer by seeing off Sloane Stephens 5-7 6-2 6-3.

Kerber revealed she overcame having to sleep in Flushing Meadows on Wednesday night as Storm Ida caused her match to be cancelled late on.

She said: “I mean we got cancelled really late. I thought that we can go straight to the hotel. But it was impossible because of the storm and the rain and everything.

“We stayed here until 3:30, 4am. I was trying to get a little bit of sleep because I had the match yesterday, just like few hours later. Yeah, that was my Wednesday night in the gym.”

Leylah Fernandez made it a night to remember for the 18-year-olds as she dumped out defending champion Osaka.

Arriving on Arthur Ashe Stadium after Spaniard Carlos Alcaraz had just seen off men’s third seed Stefanos Tsitsipas, the teenager followed suit by producing a stunning upset.

Osaka has admitted she is still struggling with the mental health issues that caused her recent absence, but she served for the match in the second set, only to suffer a 15-minute capitulation that saw Fernandez fight back to win 5-7 7-6 (2) 6-4.

In a match that finished after 1am, second seed Aryna Sabalenka beat American Danielle Collins 6-3 6-3.