Konta's previous match against Halep was a classic, when they met in the 2015 WTA Wuhan and Konta progressed 6-3 3-6 7-5 and this one proved to be no different.
Third seed Halep took the opening set 6-3 and she was serving at 5-4 to win the match in the second.
However, Konta fought back and took it to a decider, winning the tie-break 9-7.
Halep produced a number of unforced errors and they dogged her in the second and third sets as Konta won the final stanza 6-2.
The Briton will now face Venus Williams for a place in the final after she beat top seed Angelique Kerber in straight sets.
American 11th seed Williams, who is seeking her fourth Miami title, survived a long first set before she completed a 7-5 6-3 victory over Germany's Kerber on a calm evening in south Florida.
Read the original article on Eurosport: Konta beats Halep to become first British woman to reach Miami semis