It was a case of “mother knows best” for British No 3 Harriet Dart as she cruised through her first-round match at the US Open for the loss of only three games.

Dart’s coach Olga Morozova is 75, and tends not to leave the UK much these days. So Dart has co-opted her mother Susie, who used to be Middlesex’s county captain, to fulfil the role during her last two tournaments on American soil.

The arrangement is working so well that Dart told reporters: “We’ve only had one argument.” But there could be complications if she replicates her run to the third round of Wimbledon earlier in the summer.

Susie has a day job as a teacher at Arnold House School in north London, and she is due to fly home on Friday in order to be ready for the start of term next week.

“She wouldn’t say she’s coach although she does have the coach’s badge,” said Dart of her mother after thrashing France’s Chloe Paquet by a 6-1, 6-2 scoreline. “She says her role is feeding balls, and throwing me balls, but there’s much more to it than that. She’s been helping me a lot.

“She played county and vets. And a couple of pro tournaments as well, but then she went to university and had me, so that changed her plans.

“She can hit with me a little bit – she can warm me up easy – and she’s played so she understands a lot of things. I think she realises also how full-on these days are and how long it is today. It’s not like you can leave the site during the day before you go on. It’s nice to have her here.”

Dart’s long vigil in the player lounge was necessitated by two apparently endless men’s matches on Court 16.

First up was British No 10 Jan Choinski, who lost in five sets to Spain’s Roberto Carballes Baena. Their four-and-a-half-hour struggle prefaced another four-setter between France’s Alexandre Muller and Australia’s Adam Walton, which ran to just over three hours.

“I am lucky to have a few people with me,” said Dart, who is also accompanied by her boyfriend Stephen Falck, her father Nick and her sister Phoebe. “And in the changing rooms we have sofas and stuff.

“You can pretty much chill but about 30 minutes before the match I thought I should get some air, because I hadn’t had any since since I had practised at 1pm which felt like a lifetime ago. My court seemed to be going the slowest out of everyone today.”

Having turned 28 in July, Dart has the benefit of significant experience in these situations. And when her match finally began, at around 7.10pm, she flew out of the blocks by winning the first four games.

Her court coverage was as fluid as usual, but there was also a new cutting edge to her serve – traditionally a weakness – which stemmed from a couple of technical tweaks that Morozova had suggested over the summer.

“I have changed a little bit on my serve back in London,” said Dart. “It is nice to see the rewards so quickly. With that I have been sore and stiff in different places but it has been for the best.”

This victory will push Dart up into the world’s top 70 for the first time, and should make her the British No 2 – overtaking Emma Raducanu – unless Raducanu can come through her own first-round match against Sofia Kenin on Tuesday evening.

Even though Morozova is at home in Wimbledon, Dart ascribes her recent form to their collaboration.  “Honestly, she is the most amazing human being,” said Dart of the Russian legend who appeared in two grand-slam finals in 1974. “She just brings a completely different perspective to what I have had previously and it is really nice to work with a female.

“It’s quite incredible what she achieved as a player, and then obviously coaching as well [with Britons such as Laura Robson as well as two big-name Russians in Svetlana Kuznetsova and Elena Dementieva]. I was really pleased that she was really open to working with me.

“I think she’s the most positive coach I’ve ever worked with,” concluded Dart, who now faces Marta Kostyuk, the 19th seed from Ukraine, in the second round. “Even if I play a really bad match she’ll find something positive. She believes in me a lot, and it makes me believe in myself.”

Broaden your horizons with award-winning British journalism. Try The Telegraph free for 3 months with unlimited access to our award-winning website, exclusive app, money-saving offers and more.