The victory over the third-seeded Halep means Konta becomes the first British woman to reach the last four at Miami and she will face the winner of Wednesday's other quarter-final between Venus Williams and German world number one Angelique Kerber.
"I'm really happy that I've come through and I am just looking forward to my next round - it's going to be a tough one," said Konta.
Halep settled well in the opening set, with her powerful forehand dominant, but Konta fought back well in the second.
The Australia-born Konta broke early to establish a 3-0 lead but the gritty Romanian clawed her way back winning the next three games while her opponent failed to make the most of her openings in what was an error-strewn contest.
Nonethless Konta stuck at her task and was able to break again when Halep was serving for the match and force a tie-break which she eventually won 9-7 to take the pair into the third game.
There was a tense discussion between Halep and her Australian coach Darren Cahill.
Konta broke in the fifth game of the deciding set and visibly growing in confidence she broke again to take a 5-2 lead and serve out for the match.
The 25-year-old's Belgian coach, Wim Fissette, previously worked with Halep and helped with Konta's strategy.
"I think we had a good game plan going into the match. Also, I had played her once previously, so I think that was also helpful for me, just so that I have felt her ball before," she said.
"But otherwise, it was also a little bit thinking on my feet as well. Certain things that can change within a match, regardless of the game plan. I think I did a good job of just adapting, yeah, and fighting," she said.
It was a disappointing result for Halep, who, in the absence of Serena Williams due to injury, must have fancied her chances of winning in Miami.
"It was a really tough match, very high level," Halep said. "I was so close to winning, I was two points away in the tiebreak, but she played very strong and deserved to win today.
"I'm just disappointed I lost a match I had in my hands. But my confidence is there, the game is there - I just need to play matches."