Robredo came away a 2-6, 7-5, 6-4, 7-6 (8-6) winner over Gasquet, a player he hadn't beaten since the fourth round of the 2007 Australian Open. He dropped his serve just once after the first set and was 3-of-7 in his own break chances.
There were no break opportunities for either player in the third set, but Robredo was able to finish the win by taking the final two points off Gasquet's serve.
Gasquet joins fifth-seeded Juan Martin del Potro on the sidelines in Melbourne. Del Potro lost in five sets to Roberto Bautista Agut early Friday morning.
Robredo's win came in the relatively cool late-night action. Earlier Friday, temperatures reached 109 degrees Fahrenheit during the final day of a grueling heat wave. Weather is still an issue, though. Play was stopped twice Friday, and roofs closed on the main courts, because of rain, which is expected Saturday and Sunday.
Three-time defending champion Novak Djokovic, in the final match Friday, beat Denis Istomin 6-3, 6-3, 7-5. Djokovic did not face a break point until he was serving for the match at 5-4 of the third set. Istomin got that break but Djokovic answered with his fifth break of the day and then served out for the win.
The decision ran Djokovic's winning streak in Melbourne to 24 matches and he's won his last 28 matches overall.
Unseeded Florian Mayer gained a spot in the fourth round with a 7-5, 6-2, 6-2 win over 20th-seeded Jerzy Janowicz. Mayer lost only 11 points on serve and saved the only break point he faced in reaching the fourth round of a Grand Slam event for the first time.
Third-seeded David Ferrer got by No. 29-seed Jeremy Chardy 6-2, 6-7 (5-7), 6-2 while making just 24 unforced errors and seventh-seeded Tomas Berdych had little trouble in a 6-4, 6-2, 6-2 win over Damir Dzumhur.
No. 9-seeded Stanislas Wawrinka advanced in a walkover after 28th-seeded Vasek Pospisil injured his back in a doubles match and was unable to play singles Friday.
Also Friday, Fabio Fognini defeated Sam Querrey 7-5, 6-4, 6-4 and 19th-seeded Kevin Anderson rallied for a 3-6, 4-6, 6-3, 7-6 (7-5), 7-5 win over Edouard Roger-Vasselin in a 3-hour, 54-minute battle in the heat of the afternoon. It was the second time in the tournament that Anderson came from two sets down to win.
The fourth-round pairings have Anderson going against Berdych while Ferrer meets Mayer in one quarter of the draw. Djokovic plays Fognini and Wawrinka faces Robredo in the other.