Coric will try to win his second tournament this season. He won his first career title in Marrakech, Morocco, last month.
He will play the quarterfinal against Dominic Thiem of Austria, who won a thriller against Grigor Dimitrov of Bulgaria 4-6, 6-4, 7-6 (9).
Murray, who lost last year's final to Djokovic in Madrid, had 28 unforced errors in a disappointing outing under the closed roof of the center court in Madrid. He was broken three consecutive times in the first set, and once in the second to give Coric a 5-3 lead that he converted to close out the match.
Murray won his only title of the year in Dubai and had started the clay season trying to regain his form following a right elbow injury that kept him out of the Miami Open and the Davis Cup quarterfinals. He lost to Albert Ramos-Vinolas in the third round in Monte Carlo after blowing a 4-0 lead in the deciding set, then was eliminated by Thiem in the semifinals of the Barcelona Open.
Second-ranked Djokovic has also struggled this season but stayed alive in Madrid with the two-set victory over Lopez.
The Serbian player was in control during his service games and broke Lopez once in each set to secure the victory. Djokovic saved the only break point he conceded to Lopez, who was playing in the tournament for a record 16th time.
"The level was definitely very high today I thought from both players," Djokovic said. "I think Feliciano played really well, especially in the second set. I wasn't winning too many points on his service games until the last one where I managed to return many balls back in play and then win the match."
Djokovic won in Doha to start the year but failed to make it past the quarterfinals in the following four tournaments he played since then.
He will next play against sixth-seeded Kei Nishikori of Japan, who defeated David Ferrer 6-4, 6-3 to deny the Spaniard his 700th career win.
A finalist in Madrid in 2014, Nishikori is playing in his first tournament since March. He withdrew from Barcelona last month because of a right wrist injury.
Nadal picked up his tour-leading 31st victory this year by easily defeating 16th-seeded Kyrgios, who never threatened after losing the first set.
"I played a lot better," said Nadal, who struggled in a three-set win over Fabio Fognini on Wednesday. "It was a big step forward."
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.
The other quarterfinal will be played between Pablo Cuevas of Uruguay, who defeated Benoit Paire of France 7-5, 0-6, 6-1, and Alexander Zverev of Germany, who beat Tomas Berdych of the Czech Republic 6-4, 6-4.
On the women's side, third-seeded Simona Halep of Romania reached the semifinals by beating Coco Vandeweghe of the U.S. 6-1, 6-1 in less than an hour.
"I was a little bit nervous, to be honest, before the match, because she has a big serve," Halep said. "It's always tough to play with someone who has a big serve."
Halep will play Anastasija Sevastova of Latvia, who defeated Kiki Bertens of the Netherlands 6-3, 6-3.
Kristina Mladenovic of France advanced to the final four by overcoming seven double faults to beat Sorana Cirstea of Romania 6-4, 6-4. Mladenovic will play eighth-seeded Svetlana Kuznetsova of Russia, who easily defeated Canadian Eugenie Bouchard 6-4, 6-0.
Bouchard ousted Maria Sharapova in the second round.
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