The second seed rallied back from a dismal opening set to post a 1-6 7-6(4) 6-2 win over the Frenchman, seeded sixth at the tournament in Mexico.
Murray had struggled in the opening set against his first-round opponent Pablo Andujar, losing the first five games of that match, and the Scot was again slow off the mark against Simon.
The world number seven managed just one game in the first set and was only marginally more successful in the second, falling behind 2-4 and then 3-5 as Simon closed in on victory.
But Murray dug deep into his reserves to break back and then claim the set on a tie-break.
The third finally saw Murray back to something resembling his best as he swept aside Simon to open up a 3-0 lead that he never looked back from.
"In all the matches so far this week, I have not started very well," the world number seven said, conceding he did not know what was to blame for his tardy openings.
"I lost my first service game in every match and that has not helped because my opponents have gained confidence and played very good tennis.
"It's not easy starting every match from behind. I managed to turn the game at the end of the second set, started to play a little bit better. I was making sure I was not making so many unforced errors," the Scot added.
"I found a way to win the match even though I was feeling I was not playing my best tennis. I'm glad that I managed to do that.
"Maybe I just try and warm-up better, maybe practice a little bit closer to the matches."
Murray, in his first semi-final since undergoing back surgery last September, next plays Grigor Dimitrov, who needed two hours and 40 minutes to battle past Ernests Gulbis 4-6 7-6 (7-2) 7-5 in the final quarter-final match of the day on Cancha Central.
Earlier, top seed and three times winner in Acapulco David Ferrer was forced to retire with a muscle strain in his match with Kevin Anderson.
The Spaniard had won the first set 6-2 before Anderson rallied in the second and the South African was leading 4-2 when Ferrer called a halt to proceedings.
Anderson will play Alexandr Dolgopolov for a place in the final after the Ukrainian beat Croat Ivo Karlovic 6-4 7-6(4).
Acapulco International results
2-Andy Murray (Britain) beat 6-Gilles Simon (France) 1-6 7-6(4) 6-2
4-Grigor Dimitrov (Bulgaria) beat 7-Ernests Gulbis (Latvia) 4-6 7-6(2) 7-5
5-Kevin Anderson (South Africa) beat 1-David Ferrer (Spain) 2-6 4-2 (Ferrer retired)