Yorkshireman Edmund will now play French 15th seed Gael Monfils, who beat Daniel Brands of Germany 6-3, 7-5, 6-4, in round two, while Alex Ward is set to see his ranking improve by around 300 places.
Edmund also revealed he had not studied Monfils much in the past.
"Because you're British, it's the one tournament you want to do well in more than any other tournament".
Edmund said: "I always look out for them". To actually say that I've won one [a match] now is nice in terms of a professional tennis point of view.
"I would have been disappointed, but I'm not going to cry over it".
"It's nice to just win in general", he said. "It's one round out of seven, which is the reality".
Each had their own back story too with Ward having come through qualifying to reach the main draw while Edmund, without a win on grass this year, split with coach Ryan Jones last month.
"I'm completely devastated this moment", she said. From the very first ball I felt comfortable out there.
"From knowing him throughout the years though, he's obviously a very good athlete, moves well and makes a lot of balls".
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"I've played Brits in the past".
It wasn't quite the Battle of Britain some had hoped for but Kyle Edmund was still made to work harder than the rankings might have expected before claiming his first ever Wimbledon win as he saw off the qualifier Alexander Ward 4-6, 6-3, 6-2, 6-1 on Tuesday.
The 30-year-old, who helped Great Britain to a memorable Davis Cup win in 2015, was playing just his fifth match since suffering a serious knee problem in September but went down 6-4 6-4 6-3 to Baghdatis on Court 18. I played Andy a couple of times.
"But I have asked Andy some stuff before about opponents, but not every time".
"Once you go on the court, you have to get on with it and play for yourself".
"That's the main thing at the moment, just to try and stay healthy and play as many tournaments I can until the end of the year".
British number six Brydan Klein twisted his knee in a nasty fall during his first-round defeat to Yuichi Sugita.
Katie Boulter went close to a shock debut victory, but ultimately came up short against American Christina McHale. SSE provide access to exclusive sport and entertainment reward offers and they're giving customers the chance to win incredible experience packages to this year's already sold out event.
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