Japan's Kei Nishikori also reached the last eight by pulling off the first major upset of the tournament, toppling fourth-seeded Spaniard David Ferrer 7-6 (9-7) 2-6 7-6 (11-9) in a pulsating three-hour battle.
While the day got off to a gloomy start with early morning showers delaying the start of play by 30 minutes, tennis fans were treated to lineup of sparkling action with nine of the world's top 10 men battling for spots in the quarters.
Djokovic had only two break points the entire match and that was all he needed converting both.
The six-times grand slam winner went to work under a sunny South Florida sky and wasted no time claiming the only break of the opening set to go up 3-2 on way to winning the first set.
Robredo and the world number two continued to slug it out in the second but it was the Serb once again recording the only break at 6-5 then holding serve to close out the match.
"Very solid win," said Djokovic. "I just was telling myself to stay mentally tough out there and composed and not get carried away by few points.
"I have done well. I have served well. I made him play an extra shot in important moments and that's why I'm satisfied with the overall match today."
It was the second match of the tournament for Djokovic, who received an opening-round bye and advanced to the last 16 on a walkover when Florian Mayer withdrew with a groin injury, and sets up a potential mouth-watering quarter-final clash with defending champion Andy Murray.
The sixth seeded Scotsman, contesting his first event since splitting with coach Ivan Lendl last week, faces Frenchman Jo-Wilfried Tsonga later on Tuesday.
Nishikori, who had never been past the fourth round in Miami, entered new territory after out-lasting Ferrer, one of the fittest players on the ATP Tour and a finalist here last year.
The roller-coaster contest began with a first set that featured three breaks of serve by each player before Nishikori won it in a tiebreak.
Ferrer hit back by breaking the 20th-seeded Japanese to open the second and again to go up 4-1 on way to leveling the match.
The up-and-down battle continued into the third with the two men exchanging a pair of early breaks before holding serve to the finish, which ended in another gripping tiebreak and Nishikori again coming out on top.
"I was ready for longer rallies, I didn't think it was going to be this long a match but to beat him in three sets that shows me confidence I have," said Nishikori. "To go to quarter-finals here the first time is what I want to do at these Masters and big tournaments and hopefully I can keep going."