The 23-year-old from Thornhill, Ont., will next take on Australian Open champion Stanislas Wawrinka, who advanced on a walkover when Spanish opponent Nicolas Almagro retired before their match with a foot injury.
Wawrinka, the third seed, has beaten the eighth-seeded Raonic in both of their previous meetings.
"I've struggled in the past against Stan but it will be quite different this time around," said Raonic. "I'm doing a lot of things well and I'm looking forward to that challenge."
Raonic had seven aces in his third-round match and broke Robredo's serve once per set while never facing a challenge on his own serve. The Canadian also beat Robredo in their previous meeting last spring in Barcelona.
His next match marks a breakthrough as it will be Raonic's first quarter-final appearance at a clay-court 1000 Series event.
"This is a first, that's for sure," he said. "It's a result of a lot of the good things I'm doing. But I can do better and I'm looking forward to bringing that out in myself.
"This is the closest I've gotten to a big clay opportunity, so it's definitely a positive."
In second-round men's doubles play, Toronto's Daniel Nestor and Serbian partner Nenad Zimonjic defeated the Polish duo of Mariusz Fyrstenberg and Marcin Matowski 6-7(6), 7-6 (3), 10-7.
Raonic claimed the opening set against Robredo with a break in the final game, smashing down an overhead winner on his second set-point opportunity.
The Canadian jumped out to a 2-0 lead in the second set when a Robredo return wide down the line. Raonic then stayed the course to serve out the victory, finishing in just over an hour.
"I'm very pleased with myself, I showed a lot of discipline," Raonic said. "I struggled in the beginning but was able to figure it all out towards the end of the set. I gave myself a lot of opportunities afterwards."
Raonic credited his world-class serve with keeping him out of trouble.
"Serving well was a big thing today," he said. "I was also moving well and able to dictate with my forehand."
The victory boosted Raonic's record to 10-3 on the season and has given him a boost of confidence on the slower surface.
"I've played a lot of the Spanish guys in practice matches," said Raonic, the former pupil of Spaniard Galo Blanco. "I definitely have more of an understanding of what I need to do and what I can expect from them as well."
Also Thursday, defending champion Novak Djokovic raced into the quarter-finals by dispatching Pablo Carreno Busta of Spain 6-0, 6-1 in 47 minutes.
Eight-time champion Rafael Nadal also had little trouble, beating Italian Andreas Seppi 6-1, 6-3, while fourth-seeded Roger Federer shook off a slow start in a 6-4, 6-1 win against Czech player Lukas Rosol.
The second-seeded Djokovic won his second-round opener in 45 minutes and has yet to be tested this week. He next faces Spaniard Guillermo Garcia-Lopez. The top-seeded Nadal has looked sloppy at times.
"Lost only two games in the first two matches (here). Can't be better than this, for sure," Djokovic said. "I am confident on the court. I'm coming off a great American hard-court season."
Djokovic sealed the 22-minute first set with a crisp forehand. Carreno Busta received loud cheers, clenched his fists and grinned broadly when he won his only game for 3-1 in the second. Djokovic served out, tormenting him one last time with an exquisite drop shot.
He is aiming for his fifth straight Masters win. He won at Indian Wells and Key Biscayne, dominating Nadal in the latter final to record his third straight win against him in straight sets.
"I've had more time comparing to last year to practice on the clay courts and to get my game to the level where I want it to be," Djokovic said.
Nadal achieved his 300th win on clay in beating Seppi. He won this title for eight straight years until last year when he lost the final to Djokovic.
"Today I played a little bit better than yesterday," Nadal said. "I was playing at a good level."
He next faces sixth-seeded David Ferrer, who easily beat Grigor Dimitrov of Bulgaria 6-4, 6-2, while Federer will face ninth-seeded Jo-Wilfried Tsonga.
Nadal has a 21-5 career record against countryman Ferrer, winning 10 of the past 11 meetings.
"David is a tough, tough player on any surface," Nadal said. "But here on clay, always a big challenge."
Rosol, meanwhile, broke Federer in the third game and held for 3-1.
"It was a bit rocky in the beginning," Federer said.
He broke back with a stinging cross-court forehand at Rosol's feet, and then broke him to love. He clinched the set when Rosol's forehand sailed long.
Federer has a 10-4 record against Tsonga, who beat him in the French Open quarter-finals last year and in the Wimbledon quarters in 2011.
"I've seen Jo play different kind of quality matches lately, so not quite sure he's going to play, how aggressive, how passive," Federer said.
Tsonga won 5-7, 6-3, 6-0 against Fabio Fognini of Italy, who did his best to spoil the Frenchman's 29th birthday with his wild antics, including smashing rackets and shouting loudly.
In the day's only upset, Garcia-Lopez beat fifth-seeded Tomas Berdych of the Czech Republic 4-6, 6-3, 6-1.
The only other time Garcia-Lopez reached a Masters last eight was in Shanghai four years ago, when he lost to Djokovic, who leads 5-0 in their head-to-heads.
With files from The Associated Press.