The Briton played poorly in the first set, losing three service games and looking increasingly frustrated.
Coric also won his first ATP title, on clay, this year, beating the likes of Albert Ramos-Vinolas, Diego Schwartzman and Philipp Kohlschreiber en route to success in Marrakech.
"I started the match OK but when I started to go behind I didn't find any way to improve my game or to make it more hard for him, I just let the same things keep happening". If a gameplan isn't working it's not my coach's fault because I always go over the gameplan.
Despite the pressure of serving for the match, Coric remained cool to seal victory in just under 90 minutes on court when Murray fired wide to set up a quarter-final clash with Thiem.
"It's not always the worst thing losing a match, but it's sometimes the manner of how you lose the match which can be concerning or disappointing".
"I should've tried to do more about that and I didn't and that's the most disappointing thing for me".
The result is a further blow to Murray's build-up for the French Open later this month.
"I definitely think I need to be concerned about today", Murray said.
World number two Novak Djokovic came through a tricky second set against Spain's Feliciano Lopez to book his place in the quarter-finals.
The poor weather in Madrid led to the roof at the Manolo Santana court being closed which took some of the effectiveness off Lopez's powerful serve, giving Djokovic an advantage in a tournament held at altitude where the ball is considered to fly faster than at other events.
Murray has struggled to maintain the consistency that saw him dethrone Novak Djokovic at the top of the rankings at the back end of last year and a 6-3 6-3 defeat to the 20-year-old Croatian provided his latest setback.
Crowd-favourite Rafael Nadal, the only remaining Spaniard in the tournament, plays Australian Nick Kyrgios later on Thursday.
However, Djokovic salvaged the game and then immediately broke the Spaniard's serve to set up a meeting with Japan's Kei Nishikori, who swept aside another home favourite David Ferrer 6-4 6-3.