It was only the second set Bacsinszky had dropped during the tournament, and the 19th-seeded Swiss made sure it would be her last of the day as she prevailed in a match that stretched nearly two hours and 30 minutes.
Watched by a crowd that included Swiss ski champion Lara Gut, Bacsinszky ended the match in style with a cross-court backhand winner.
"I came back from a long road," 26-year-old, who has battled back from injury, said in a courtside interview.
"For the past year she (Halep) has been an inspiration for me because I believe in hard work and I know she's one of the players which work really hard. To be able to play that well against her and catch the win is really amazing."
Halep's exit left the event with only one of the top 12 women's seeds - number two Angelique Kerber, the Australian Open champion.
Bacsinszky will meet Russian Sevtlana Kuznetsova in the semi-finals after the 15th seed beat compatriot Ekaterina Makarova 6-7(3) 6-4 6-3.
Kuznetsova shocked world number one Serena Williams in straight sets a day earlier, and is playing past the third round of a tournament for the first time since January.
She dropped the first set but battled back in a topsy-turvy second before breaking Makarova three times in the decider.
Bacsinszky has taken down some big names en route to the semi-final, disposing of third seed Agnieszka Radwanska and 16th seed Ana Ivanovic in previous rounds.
Her Miami form follows on from a stellar 2015 campaign during which she cracked the world top 10 for the first time and won two tournaments before a knee injury prematurely halted her campaign in October.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Larry Fine)
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