Federer received a standing ovation after closing out the match with a serve-and-volley point, hitting a backhand volley that Robredo couldn't handle.
The win avenged Federer's straight-set loss to Robredo in their last meeting in the fourth round of the U.S. Open. Apart from that defeat, Federer has won all of his 11 matches against the Spaniard.
Federer has lost only 32 games in four matches at the All England Club, the same number as defending champion Andy Murray.
"Clearly I'm very pleased with the first week, and here we go now into the quarters," Federer said. "It's always really exciting being so deep into a tournament and feeling you are closer to the finish line.
"I've played a lot of matches so things are exactly where I want them to be, but then again you're sort of only in the quarterfinals and that's when the tournament kind of really starts."
Federer's win set up an all-Swiss duel with Australian Open champion Stan Wawrinka, who beat Feliciano Lopez 7-6 (5), 7-6 (7), 6-3 to make it to the Wimbledon quarterfinals for the first time.
Federer holds a 13-2 record against Wawrinka, though Wawrinka won their last encounter in the Monte Carlo final in April.
In women's play, French Open runner-up Simona Halep breezed into the quarterfinals with a 6-3, 6-0 win over Zarina Diyas of Kazakhstan.
The third-seeded Romanian needed just 57 minutes to beat the 72nd-ranked Diyas, who was playing at Wimbledon for the first time.
Her next opponent will be 2013 finalist Sabine Lisicki of Germany, who overcame shoulder trouble to defeat Yaroslava Shvedova of Kazakhstan 6-3, 3-6, 6-4 in two hours.
Halep is the second Romanian woman to reach the quarterfinals at the All England Club, joining Virginia Ruzici, who made it to the final eight in 1978 and 1981. Ruzici is now Halep's manager.
Halep, who has reached the quarterfinals at all three Grand Slams this year, is the only player among the top four seeds left in the draw. No. 1 Serena Williams, No. 2 Li Na and No. 4 Agnieszka Radwanska are all out.
The 19th-seeded Lisicki took an injury timeout while facing break point at 1-1 in the second set and was treated by a trainer on her right shoulder.
After returning to the court, Lisicki saved the break point and held serve, then broke in the next game for a 3-1 lead. She was broken by the 72nd-ranked Shvedova while serving for the match at 5-3, but broke again in the next game to close it out. An emotional Lisicki celebrated by falling to her knees at the baseline and resting her forehead on the grass.
Both players piled up more unforced errors than winners. Lisicki had 33 errors and 29 winners, while Shvedova had 34 and 18.