"Just looked at the stats ... 24 unforced errors is far too many. That's a set's worth. That's where half the points went," a despondent Murray said. "I hope that's not going to be the case over the next five, six weeks. I've had tough losses on the clay before and I've come back well from them. Hopefully today will be the same."
Murray won less than half of his first-serve points and his touch was off throughout the match.
Wawrinka said Murray's negative body language was easy to read when he's not playing well on clay.
"I think when he's not feeling that good, he cannot just play from the baseline against me. For me it's perfect because he's playing so slow," Wawrinka said. "You never expect to beat a top-four player, especially that easy."
Sixth-seeded Jo-Wilfried Tsonga also advanced with a clinical 6-3, 6-0 win against Jurgen Melzer and fourth-seeded Tomas Berdych lost 6-4, 6-2 to Fabio Fognini of Italy.
The third-seeded Nadal comfortably dispatched Kohlschreiber without facing a single break point. The Spaniard next faces Bulgaria's Grigor Dimitrov.
"He's playing well," Nadal said. "He's able to play very aggressive, very good technique. He's (a) complete player."
Nadal picked off Kohlschreiber from the back of the court, clipping a forehand winner past him to break and then holding to love to move 5-3 up in the second set. He served out, clinching victory when the German scooped a backhand into the net.
Djokovic did a skip of delight when Monaco sent a return long on match point. The Serb twisted his right ankle on Davis Cup duty against the United States a little less than two weeks ago and had been doubtful to play at Monte Carlo, where he lost to Nadal in last year's final.
"I am trying to take things day by day," he said. "I didn't know what to expect today and I'm really glad that I won."
Djokovic had a lengthy chat with a trainer at the end of the first set and decided to carry on playing. He went up a break early in the second set and held for 3-0.
He plays Jarkko Nieminen in the quarterfinals after the Finn beat fifth-seeded Juan Martin del Potro of Argentina 6-4, 4-6, 7-6 (4).
Nieminen sealed victory on his second match point when del Potro sent a big forehand wide. Both players dropped serve five times in a gritty contest lasting 2 hours, 37 minutes.
Tsonga did not concede a single break point and took a little under one hour to win, setting up a quarterfinal against Wawrinka.
"He's playing really fast," Wawrinka said. "He's a really, really good clay-court player."
Dimitrov beat Florian Mayer of Germany 6-2, 6-4 and Nadal was wary of him.
"I have to try to be solid all the time and put him in not easy positions to attack the ball," Nadal said.
Fognini next plays seventh-seeded Richard Gasquet after he beat No. 9 Marin Cilic 7-5, 6-4.
-AP