Federer, 32, will now face fellow Swiss ace Stan Wawrinka.
Wawrinka, who won his first grand slam at the 2014 Australian Open, defeated Spain's Feliciano Lopez in straight sets.
"It will be amazing to play Stan in the quarterfinals here, hopefully on Centre Court, because things like that don't happen too often in Swiss sport," Federer told the BBC.
"But I'm pleased with the way I am playing. I'm serving and moving well and all the things are happening that need to be happening to go deep in the tournament."
Wawrinka, now the Swiss No. 1 after many years in Federer's shadow, claimed a 7-6 7-6 6-3 win to reach his second Wimbledon quarterfinal.
Federer needed just 21 minutes to win the first set 6-1 against Robredo before wrapping up the second 6-4.
Robredo, seeded 23, simply had no answer and Federer closed out the match by taking the third 6-4.
In the women's draw, last year's finalist Sabine Lisicki overcame a scare against Yaroslava Shvedova of Kazakhstan.
The German prevailed 6-3 3-6 6-4 to set up a last eight tie with number three seed Simona Halep of Romania.
Halep, the highest ranked player remaining in the draw, Zarina Diyas of Kazakhstan 6-3 6-0.
"I'm not surprised that I play a quarterfinal tomorrow," Halep told reporters.
"I'm third in the rankings so I have to be here in the quarterfinals. I've played really well and I have confidence in myself.
"I deserve to be here because I'm working hard every day and it's normal to be in the last matches of tournaments and grand slams."
In the women's doubles, Serena Williams was forced to retire from her match with a viral illness.
Williams, playing alongside sister Venus, required lengthy treatment from the doctor and trainer before the game against Switzerland's Stefanie Voegele and Germany's Kristina Barrois.
She decided to play following the warm-up but in her first service game, she served four double faults and retired with the score at 0-3.
In a statement issued by the WTA Tour, Williams said: "I am heartbroken I'm not able to continue in the tournament. I thought I could rally this morning, because I really wanted to compete, but this bug just got the best of me.
"I want to thank my sister, Kristina, Stefanie and our teams for their support. We were all looking forward to a great match.
"From the bottom of my heart, I thank all of the fans for their cheers and understanding. I look forward to returning to Wimbledon next year."