Halep, the tournament's third seed, broke early on in the match in the fourth game to take the opening set, despite Konta saving set points at 5-2 down.
The Brit battled back but was on the verge of being knocked out of the competition when - after breaking Halep in the 10th game of the second set to force a tie-break - she found herself 5-3 down.
However, three points in quick succession swung things back in her favour and she levelled up the match 9-7 in the tie break.
In the deciding set, Konta broke early on and saw off the challenge from Halep to become the first British woman to ever reach the last four at the Miami Open - where she will face top seed Angelique Kerber or Venus Williams.
The other semi-final will be between Caroline Wozniacki and Karolina Pliskova.