• Defending champion wins 4-6, 7-5, 6-2 in second round • ‘I wasn’t playing my best and I just had to find a way’

The bad news for the champions of tennis is that they can no longer take anything for granted. The good news for the future of the game is that the insurrection springs from a perceptible rise in standards deep in the rankings – as the title-holder, Sloane Stephens, was reminded when she narrowly avoided following the world No 1 Simona Halep out of the US Open on day three.

Suffering on another day of intense heat, the American struggled at first to douse the impertinent skills of an opponent many watching on Arthur Ashe Court would have known little about, Anhelina Kalinina, who came to the second round as a qualifier ranked 134 in the world and left it exhausted but $93,000 (£72,000) richer.

Stephens was relieved more than ecstatic after two hours and 46 minutes of toil that brought her an edgy 4-6, 7-5, 6-2 win. “I wasn’t playing my best and I just had to find a way,” she said courtside, as the loser headed for the sanctuary of the locker room. “Not ideal conditions, but just happy to be through. I never gave up, fought my tail off. Hope the next one’s at night.”

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She maintained the ludicrous facade of not wanting to know who her opponent would be on Friday, but will be more alert to the dangers when she sees Victoria Azarenka across the net, a much-revived former contender who flew past the dangerous Australian Daria Gavrilova in two sets for the loss of three games in 82 minutes.

Kalinina, meanwhile, had already earned nearly half of her career prize money by getting out of the first round when her opponent, Kathinka Von Deichmann, became one of five casualties of the heat that has gripped this tournament like a glove. But the 21-year-old Ukrainian rode her fortune stylishly on Wednesday.

Three years ago she was another promising teenager, lurking outside the top 150. She broke through that barrier only to spend 10 months out with a shoulder injury, and returned to rebuild her career in fringe tournaments. Here, though, all the promise flowered again in a first set of 61 minutes, as she busted up her bamboozled and strangely hesitant opponent.

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Stephens hit back hard to go 3-0 up in the second – only to need wrapping for a blister on the second finger of her right hand. When she hit long to surrender a break lead in the 10th game, she had only herself to blame for not challenging Kalinina’s previous winner, which plainly hit the tramlines.

From seeming comfort, Stephens was hurled back into uncertainty, but she repaired the damage quickly, breaking back to love, before holding through deuce to force a third set. The players took their 10-minute heat break then traded on roughly equal terms for 20 minutes before weariness kicked in savagely for Kalinina. A double fault and a wayward backhand handed a second break to Stephens, who looked solid again at the end.