Murray worked out of danger after dropping the first set to Nishikori with the 30-year-old finding a way. The world number one fell behind two breaks and the eighth seed served out the set after just 35 minutes of play.
The turning point seemed to come at deuce in the third game of the second set, when Murray caught his ball toss and was sanctioned by umpire Carlos Ramos for taking too long between serves.
Stan Wawrinka is coming off a straight sets win over Marin Cilic that took 25 games to decide. Only Bacsinszky has been to the French Open semifinals in the past, losing to Serena Williams at that stage in 2015.
But Murray, now just two games away from his third different Grand Slam trophy, battled back in his typical, gritty fashion - winning five games in a row to take the second set and level up the match. All four players are seeking a first grand slam title. "It was windy, so the timing was hard, but once I was able to get into a rhythm and keep him away from the baseline, that made the difference", said Murray. He's a great player.
"The draw looked fairly tough in the beginning but now you would have to say you would be somewhat surprised if he didn't get at least to the semis, if not the final", McEnroe said.
On the clay courts of Paris, determined the semifinalists of the 116th version of the Open championship of France.
The 32-year-old is the oldest man to make the last four at Roland Garros since Jimmy Connors in 1985.
In the women's category, Halep defeated 21st-seeded C. Suarez Navarro of Spain 6-1, 6-1 and Garcia defeated fellow compatriot Alize Cornet 6-2, 6-4 in an all-French affair again on Philippe-Chatrier.
Nishikori surpassed Jiro Sato, who died in 1934, for the most quarterfinal appearances by a Japanese man, as he made the last eight for the second time in three years in Paris.
In the quarterfinals, Nadal will face Pablo Carreno Busta, while Djokovic meets Dominic Thiem.