However, the second seeded Frenchman had to work hard before finally digging out a 6-7 (7) 6-4 7-6 (3) victory.
It looked as if Gasquet would have to wait another week for his landmark win when he fell 5-2 down in the deciding set but he battled back for the win.
Top seed David Ferrer had an easier time of things as he beat Tobias Kamke of Germany 6-3 6-2.
Former world number three Nikolay Davydenko and former world number seven Gael Monfils showed they can both still produce with victories over seeds.
Davydenko beat fourth seed and fellow Russian Mikhail Youzhny 7-5 6-3, while Monfils took care of third seeded German Philipp Kohlschreiber 6-4 2-6 6-4.
Sixth seed Viktor Troicki also lost to Lukas Lacko of Slovakia.
At the ATP event in Brisbane, third seed Gilles Simon came through his second-round match against Alejandro Falla 7-6(5) 7-6(5).
The Frenchman had a first serve percentage of just 48, but took both breakers 7-5 to overcome the world number 54.
In temperatures as high as 35 degrees on one of the Queensland Tennis Centre's outside courts Simon, a semi-finalist at the event last year, struggled through a testing encounter in conditions similar to those in which he struggled at last year's Australian Open.
"It was very hot, even with the roof, and it was very difficult playing today," Simon said. "But that's good, that's why I came here. It's such good preparation for Melbourne.
"This year I want to do my best to get my body ready for the Open. In the off season I didn't work on my game at all because I know that's in good shape. I just wanted to tune up physically."
After beating the Colombian, Simon will next take on Marcos Baghdatis in the quarter-finals, with the Cypriot having defeated sixth seed Florian Mayer.
Top seed Andy Murray is in second-round action against Australian wildcard and world number 199 John Millman on Thursday, with a potential quarter-final contest with former world number one Lleyton Hewitt on the cards.
In Chennai, second seeded Serb Janko Tipsarevic eased past Edouard Roger-Vasselin of France to set up a third-round, quarter-final encounter with Go Soeda.
Japan's Soeda, seeded eighth in India, took three sets to overcome home hope Prakash Amritraj, while fourth seed Stanislas Wawrinka secured a regulation win over Cedrik-Marcel Stebe.
The Swiss next meets Slovenia's Aljaz Bedene, who registered the shock of the day with a straight set win over sixth seed Robin Haase.
3-Gilles Simon (France) beat Alejandro Falla (Colombia) 7-6(5) 7-6(5)
4-Alexandr Dolgopolov (Ukraine) beat Jarkko Nieminen (Finland) 6-2 4-1 (Nieminen retired)
5-Kei Nishikori (Japan) beat Tommy Robredo (Spain) 6-3 6-3
Marcos Baghdatis (Cyprus) beat 6-Florian Mayer (Germany) 6-4 6-2
1-David Ferrer (Spain) beat Tobias Kamke (Germany) 6-3 6-2
2-Richard Gasquet (France) beat Grega Zemlja (Slovenia) 6-7(7) 6-4 7-6(3)
Gael Monfils (France) beat 3-Philipp Kohlschreiber (Germany) 6-4 2-6 6-4
Simone Bolelli (Italy) beat Lukasz Kubot (Poland) 2-6 6-4 6-4
Lukas Lacko (Slovakia) beat 6-Viktor Troicki (Serbia) 6-3 7-5
Nikolay Davydenko (Russia) beat 4-Mikhail Youzhny (Russia) 7-5 6-3
Daniel Brands (Germany) beat Mohamed Safwat (Egypt) 4-6 6-3 6-2
Paolo Lorenzi (Italy) beat Lukas Rosol (Czech Republic) 6-0 6-4
4-Stanislas Wawrinka (Switzerland) beat Cedrik-Marcel Stebe (Germany) 6-4 6-3
Aljaz Bedene (Slovenia) beat 6-Robin Haase (Netherlands) 7-5 6-3
8-Go Soeda (Japan) beat Prakash Amritraj (India) 7-6(2) 3-6 6-4
2-Janko Tipsarevic (Serbia) beat Edouard Roger-Vasselin (France) 6-2 6-3