Ferrer, the world No. 5, overcame some rustiness and managed to post a 5-7, 6-3, 6-2 victory over German qualifier Dustin Brown.
He'll next play Germany's Tobias Kamke, who whipped Spaniard Ruben Ramirez Hidalgo 6-1, 6-2 in another first-round match.
World No 10, Gasquet, meanwhile, dispatched Czech qualifier Jan Hernych in straight sets 6-3, 6-4 at the Khalifa International Tennis Complex.
He now goes on to face Slovenia's Grega Zemlja, who earlier doused Italy's Fabio Fognini 6-2, 3-6, 6-4.
Also Tuesday, fourth-seeded Russian Mikhail Youzhny pulled out his 400th ATP Tour match win with a 4-6, 7-6 (7-3), 6-1 triumph over Germany's Benjamin Becker, while Frenchman Gael Monfils -- last year's runner-up -- bested Qatari wild card Mousa Shanan Zayed 6-0, 6-3.
Not doing as well was fifth-seeded Frenchman Jeremy Chardy, who was bounced by German qualifier Daniel Brands 6-4, 6-4.
Another mild upset came when eighth-seeded Spaniard Pablo Andujar was done in by Czech Lukas Rosol 6-2, 7-6 (7-2).
Slovak Lukas Lacko and Egyptian Mohamed Safwat also posted first-round wins on Day 2.
Wisconsin business offering 'therapeutic cuddling' forced to close
Megyn Kelly: Santa Claus and Jesus are both white men