He will face either Italian third seed Andreas Seppi or Spanish No. 8 Marcel Granollers.
Gasquet, the No. 17, will face Kazakhstan's Andrey Golubev after beating Italy's Andreas Seppi 6-3 3-6 6-3 6-1.
CNN: Champion Djokovic backs calls for change in men's tennis
No. 2 Andy Murray, the champion in 2009 and runner-up last year, beat No. 16 Andreas Seppi 6-2, 6-4.
There were also wins for France's Jo-Wilfried Tsonga, Italy's Andreas Seppi, Belgium's David Goffin and eighth seed Richard Gasquet.
The 32-year-old wrapped up a straightforward 6-3 6-4 victory over Italy's Andreas Seppi.
The world No. 36 came from one set down to beat Italian 21st seed Andreas Seppi 7-5 6-3 6-2 6-2.
Meanwhile, home favorite Bernard Tomic built on his shock win over world number one Novak Djokovic by defeating Andreas Seppi.
Djokovic will next face Roger Federer, who easily dispatched home hope hope Andreas Seppi 6-1 6-2 in the final match on court Friday.
The Spanish seventh seed also won his second-round match on Court Central, coming from behind to beat Italy's Andreas Seppi 6-7 (3-7) 6-4 6-4.
The 2012 U.S. Open champion breezed beyond Italy's Andreas Seppi to set up a last-eight meeting with Croatia's Marin Cilic, who upset fifth seed Jo-Wilfried Tsonga.
The German 24th seed earned a clash against Igor Andreev, with the Russian 24th seed beating unseeded Italian Andreas Seppi 6-1 7-6 (7-5) 4-6 7-6 (8-6).
Gulbis, renowned for his fiery temperament, smashed his racket during the first set of his match against Andreas Seppi as he lost five straight games to concede it 7-5.
Eighth seed Mikhail Youzhny was ousted by Germany's Florian Mayer 6-4 6-7 (5-7) 6-1 while Serbia's Marin Cilic, seeded 14th, went down 6-2 6-2 to Andreas Seppi of Italy.
He had won the title twice, most recently in 2011, while Italy's Andreas Seppi claimed the crown in 2012 in what has turned out to be its last staging.
World No. 46 Andreas Seppi beat his fellow Italian, qualifier Stephane Bohli, 6-1 7-5 to set up a second-round clash with either Serbian eighth seed Viktor Troicki or Austria's Jurgen Melzer.
"I played very badly, but I won thanks to my fighting spirit, " Djokovic said after struggling to beat Italy's world No. 25 Andreas Seppi 4-6 6-7 (5-7) 6-3 7-5 6-3 in more than four hours on Sunday.
Andreas Seppi gave Italy an early lead as he came from two sets down against Vasek Pospisil to win 5-7 4-6 6-4 6-3 6-3 in three hours and 15 minutes, but Canadian No. 1 Milos Raonic beat Fabio Fognini 6-4 7-6 (7-4) 7-5.
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