The Swiss third seed won through 7-5 4-6 7-6 (9-7) 6-2, and will face either fifth seed Milos Raonic or 10th seed Kei Nishikori in the quarter-finals at Flushing Meadows.
Wawrinka amassed 75 winners to Robredo's 19, however he did also commit 58 unforced errors to the Spaniard's 31.
Robredo, the 16th seed, was a valiant opponent for the Australian Open champion, and had chances to win each of the first three sets.
The Spanish baseliner served for the opener at 5-4 up, but was broken to 15, before another break in the 12th game saw Wawrinka complete an unlikely comeback to take the first set.
Robredo squared the match at one-set all, breaking in the fifth game of the second set before holding serve thrice in succession.
Like any good best-of-five set match at the majors, there was a tense third-set tie-break after both players held serve throughout the set.
Robredo was first to set points, leading 6-4 in the tie-break, but some unparalleled Wawrinka aggression saw him save both before taking the lead on his second set point.
The tie-break defeat clearly upset Robredo, and the pair exchanged words early in the fourth set following a successful Wawrinka HawkEye challenge as the match grew tense.
The Swiss had seemingly crushed Robredo's spirit, and the Spaniard was broken in the first and fifth games of the fourth set - as Wawrinka signposted the latter with his now-trademark index-finger-to-temple celebration.
That mental edge saw Wawrinka then serve out the remainder with little trouble, winning eight of eight points on his own serve thereafter as he entered the last eight for consecutive years in New York.