The Scot was near-faultless versus 31-year-old Robredo during his 6-2 6-4 7-5 win, which maintains his immaculate record of not dropping a set at the Championships.
Murray can now look forward to a two-day break at SW19 before his Monday match, when he will face either Mikhail Youzhny or Viktor Troicki.
Thirty-second seed Robredo had never progressed to the final 16 but, at this strangest of Wimbledons, anything was possible, and Robredo has recent experience of making the seemingly impossible possible.
Since returning last summer from surgery on a hamstring problem that had dogged him for a long time, Robredo has surged back up the rankings, reaching 29th prior to this tournament.
At the French Open, he won three consecutive matches from two sets down, the first man to achieve such a feat at a grand slam for 86 years.
And despite a solid third set from Robredo, Murray made sure there were no late twists.