Konta defeated the Romanian to reach the ladies' singles semi-finals for the first time since Virginia Wade was beaten by Chris Evert, of the United States, in 1978.
With no Serena Williams or Maria Sharapova — who returned to the sport earlier this year after a drugs ban, but missed out on Wimbledon due to a muscle injury — Konta stands a fantastic chance of winning the tournament on Saturday (July 15) and she has many celebrity supporters.
Rolling Stone Mick Jagger tweeted his support for her on Wednesday.
Wade herself had won the Wimbledon title in 1977 and no British woman has got anywhere near winning it or any other Grand Slam title since the late 70s.
British tennis was in the doldrums throughout the 1980s and 1990s and Tim Henman's best effort came to nought when he lost a dramatic semi-final against Croatia's Goran Ivanisevic in 2001.
But Andy Murray, a Scotsman, burst onto the scene a decade ago and in 2013 he won the men's title, beating Serbia's Novak Djokovic and ending a wait of 77 years ago.
Murray won it again in 2016 but was knocked out by Sam Querrey, of the United States, in the men's quarter-finals on Wednesday.
But women's tennis in Britain has been struggling to find a star.
Laura Robson and Heather Watson have both flattered to deceive, but Konta, 26, is looking like she might be the real thing.
Ironically, although she is flying the Union Jack, she was born in Australia. Her parents are both Hungarian and she only moved to Britain 12 years ago.
Australian tennis reporter Leo Schlink wrote an article earlier this year in which he lamented Tennis Australia's decision to cut Konta's funding in 2004.
"Konta's funding was cut because she was regarded as lacking the requisite talent and potential," he wrote.
"Soon after, Konta and her family walked, eventually settling in England where British Tennis embraced her. The rest is history," he added.
But hungry British tennis fans have taken her to their hearts instantly and are now rooting for her to end decades of failure.
Konta fought back from a set down to win against Halep and was watched by Wade herself, who is now 72.
"I was so happy for her. I know how much pressure there is. It's wonderful to be in an atmosphere like that, and she behaves very nicely," said Wade.
On Thursday, Konta will take on Venus Williams, whose heavily pregnant sister Serena is not playing in the tournament this year.
If she wins, she will play the winner of the match between Garbine Muguruza, from Spain, and complete outsider Magdalena Rybarikova, from Slovakia.
Konta has won three out of her five previous matches with Venus and she shows a confidence, desire to win and lack of modesty which is distinctly un-British.
"I've always believed in my own ability and I've always dreamt big," she said.