The British sixth seed was locked in an intense clash with Halep on Centre Court but came through with a 6-7 (2), 7-6 (5), 6-4 victory in just over 2½ hours to reach the last four for a first time.
Britain's last female Wimbledon singles champion was Virginia Wade in 1977. Konta is the first British woman to reach the semifinals at SW19 since Wade in 1978.
"To be in the semifinals of my home slam, and to do that in front of a full Centre Court, it's pretty, pretty special," Konta said. "The level of tennis that both of us played today, it was just a tremendous match.
"Just to be a part of a match like that again. I've been very fortunate this Championships, I've had two of those now, I feel very lucky.
The victory sets up a semifinal clash with Venus Williams on Thursday, while defeat for Halep means Karolina Pliskova will overtake Angelique Kerber as the new world No. 1 when the latest WTA rankings are released Monday.
"We've had a few battles in the past," Konta added on Williams. "She's got the better of me the last time we played, so I'm really looking forward to playing her."
In a brutal, hard-hitting encounter on the London Grand Slam's main show court, Konta had to battle back from the brink to keep her title aspirations alive.
No. 2 seed Halep came within two points of reaching the last four when serving at 5-4 in the second set tiebreak, only for world No. 7 Konta to hit back and force the decider.
There appeared to be some animosity before the start of the third set when Konta left her opponent stewing in her chair after walking off court for a six-minute long toilet break. Halep was not amused and, after five minutes, questioned chair umpire Kader Nouni how long Konta was allowed to take.
However, the Briton remained unfazed and seized her chance in front of the home crowd.
She forced a break of serve in the fifth game of the final set, before keeping her nerve to serve out a famous victory -- although Konta's winning moment was soured slightly by a scream from the crowd during the final point.
"I heard a scream, I think it was off my side of the court, towards the left," Konta said.
"But I think the crowd, there were some overenthusiastic moments, which happens with every match when it comes to a tense moment. I think it's a part of sport, the crowd getting excited and getting sometimes a little too involved."