World number one Nadal will meet Britain's Murray after the Spaniard fought back from a set down to beat Russia's Mikhail Youzhny 6-7(4) 6-2 6-1.
Nadal had been involved in a dogfight with Frenchman Gilles Simon late on Wednesday and looked weary as Youzhny pegged him back by claiming the first set on a tiebreak and breaking early in the second.
But the 13-times grand slam winner, who has endured a mixed start to the claycourt season by his own high standards, turned on the after burners, winning 11 straight games to take the match away from his opponent.
Murray celebrated his 27th birthday with a determined display to beat Austria's Juergen Melzer 7-6(1) 6-4.
The last time Murray and Nadal met was in a hardcourt encounter in Tokyo in 2011 in which the Briton came out on top, but the Spaniard holds the upper hand overall in their rivalry with a 13-5 win-loss record.
"It’s a positive match to play because you play against one of the best players in the world after two tough days," Nadal said ahead of Friday's match with Murray.
"I don’t know how he has changed his play. I know he has a complete game but I have to focus on myself."
STAYING POWER
Second-seeded Serbian Novak Djokovic also had to work hard at first before easing past Philipp Kohlschreiber 4-6 6-2 6-1.
His famed staying power was on show as he battled back from losing the opening set to the dogged German before blasting his way into the last eight with some superb shot making.
In the women's event, also at the Foro Italico, China's second seed Li Na snapped a career-long jinx against Australian Samantha Stosur, winning 6-3 6-1 having lost their previous six meetings, and will meets Italy's Sara Errani in the last eight.
World number one Serena Williams brushed aside fellow American Varvara Lepchenko 6-1 6-2 to draw level with sister Venus for the most career wins among active players. Serena's 652nd win set up a quarter-final against China's Zhang Shuai.
Serbia's Ana Ivanovic hit 22 winners to defeat Maria Sharapova 6-1 6-4 and next faces Spain's Carla Suarez Navarro.
The 36-year-old Haas, who reached the French Open quarter-finals last year for the first time in his career, produced some inspirational tennis to frustrate third-seeded Wawrinka.
"I just tried to focus on what I was doing well and all of a sudden it clicked," 15th seed Haas told Sky Sports. "Once you are in a third set anything can happen."
Wawrinka, champion in Monte Carlo this year and fancied for a strong run at the French Open later this month, became involved in a dispute with the umpire early in the second set and his form tailed off.
Haas plays Grigor Dimitrov in the quarter-finals after the Bulgarian upset sixth seed Tomas Berdych 6-7(3) 6-2 6-2.
Also through is big-serving Canadian Milos Raonic who saw off Frenchman Jo-Wilfried Tsonga 7-6(5) 6-4.
Scurrying fifth-seeded Spaniard David Ferrer ground down Latvia's Ernests Gulbis 6-2 6-3 to set up a quarter-final against Djokovic.
(Writing by Martyn Herman and Toby Davis, editing by Ed Osmond and Ken Ferris)
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