The Briton showed vast resolve on Centre Court to fight back for a 6-7 (2-7) 7-6 (7-5) 6-4 victory against Halep - who would have become world number one with a win - and will be the first British female since Virginia Wade in 1978 to reach the last four.
Halep, who will become the new world number one if she makes the semi-finals, next faces Britain's sixth seed Johanna Konta.
It was Halep that triumphed 7-2 in the first tie-break as the Romanian claimed a one-set advantage, but Konta fought back to take the second tie-break 7-5 as the quarter-final clash headed for a third set.
"It's very well rehearsed and I'm ultra-impressed, I haven't seen a player really with the same sort of dedication and determination for a long time".
Halep looked at the umpire, seemingly hoping to have the point replayed.
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The British No 1 has won 11 ITF titles and four ITF doubles titles during her career won her first WTA title on the main Tour last July and collected her second in Sydney this January.
"I'm playing against another tennis player, another opponent", said Konta.
Halep, who has a chance to take over as the top-ranked player this week, will next face Johanna Konta.
"She finished past year extremely well and kind of showed that she had the potential to get there with the wins that she was having", Murray said.
The 20-year-old idolised Venus's sister Serena Williams, the reigning champion who is absent while she prepares to have her first child, and has modelled her go-for-broke game after the American legend.
In the opening seven days of play at this year's competition, 14 matches on Centre Court have been from the men's singles draw, with just eight coming from the women's.
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Konta said, after her match: "I think there's definitely an element of knowing your opponent slightly more if you've played against her before".
Garcia's father, Louis Paul, is also her coach. "I just want to go there and win it". This match against the Romanian will be her big stern test. At the French Open she was a few points from winning her first major before capitulating in the finals. "I'm not playing against a crowd and I'm not playing against a past experience", said Konta.
All 16 fourth-round matches at the All England Club are played on the same day, a schedule that will produce all eight male and all eight female quarterfinalists.
"But sometimes people can say bad things and when you're a little bit down it can bring you even more down, so you have to be careful".
Angelique Kerber through to face Garbine Muguruza in fourth round
Rogers was forced to save six break points in the very first game which she did admirably, but could not succeed a seventh time. Kerber has struggled for form in recent months and at times was blasted off court by Rogers' heavy hitting from the baseline.