The Croatian arrived at the tournament in good form following his first Association of Tennis Professionals (ATP) Tour title in Morocco in April.
And one poor service game from the Wimbledon champion cost him dear in the second as Coric served out for the match to set up a quarterfinals meeting with Dominic Thiem.
World No. 2 Novak Djokovic also reached the quarter-finals of the Madrid Open on Thursday with a straight set victory (6-4 7-5) over Feliciano Lopez.
While Kei Nishikori set up a quarter-final clash with Djokovic by sweeping aside David Ferrer 6-4, 6-3 in the third round. Murray, who won the last five tournaments of the 2016 season in an incredible run to become World No 1, now stands at a middling 4-4 from his last eight matches.
Twenty-year-old German Alexander Zverev continued his fine form after claiming his second title of the year in Munich last week as he ousted 11th seed Tomas Berdych 6-4, 6-4.
Murray needs to buckle up if he is to retain his spot at the top of ATP world rankings as the other three of the Big Four - Novak Djokovic, Roger Federer and Rafael Nadal - are progressing well to cut down the gap at the top.
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Nerves then seemed to get the better of the young Croat who made two forehand errors to allow Murray to immediately break back, but the three-time Grand Slam champion made unforced error after unforced error to hand the initiative to Coric. 'Yeah, wasn't great. I started the match okay, 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. "It's not always the worst thing losing a match, but it's sometimes the manner of how you lose the match".
"I definitely think I need to be concerned about today", said Murray. "Hopefully, I can do that, you know, with my team, and play better in Rome and Roland Garros".
One break in the sixth game was also enough for Murray to take the second set as he easily served out for the match.
She will now play Anastasija Sevastova of Latvia for a place in the final after she overcame Kiki Bertens of The Netherlands 6-3, 6-3.
The fifth-ranked Nadal will play ninth-seeded David Goffin of Belgium, who cruised to a 6-4, 6-2 win over fifth-seeded Milos Raonic of Canada.
On the women's side, third-seeded Simona Halep of Romania reached the semifinals by beating Coco Vandeweghe of the US 6-1, 6-1 in less than an hour. He withdrew from Barcelona last month because of a right wrist injury. "I was a little bit surprised but he is a great champion and he'll be back for sure".