The Slovakian beat Colombian qualifier Falla 6-3 6-2 in less than an hour to book his place alongside US Open semi-finalist Gasquet, who is ranked ninth in the world and seeded second in Thailand, while Roberto Bautista Agut progressed to meet top seed Tomas Berdych.
Switzerland's Marco Chiudinelli stood his ground against the Spaniard in Bangkok, but eventually succumbed to a 7-5 7-6 (7/2) defeat.
Japan's Go Soeda faces fifth seed Feliciano Lopez after producing a 3-6 6-1 6-2 win over Santiago Giraldo and Bernard Tomic is through to a second-round clash with Gilles Simon.
The Australian had come from behind to lead Ivo Karlovic 6-7 (3/7) 6-4 4-2 when the veteran Croatian retired hurt.
Elsewhere, a three-set battle between Lukas Rosol and Lukasz Kubot saw the former player emerge triumphant over his near namesake after two hours.
The big-hitting Czech - famous for his second-round Wimbledon ousting of Rafael Nadal in 2012 - recovered from an early wobble to win 3-6 6-3 7-6 (7/2).
Spain's Feliciano Lopez swept aside Serbian wildcard Laslo Djere in the first round of the Thailand Open in Bangkok.
The sixth seed won 6-3 6-3 in one hour 10 minutes, breaking once in the first set and twice in the second.
His compatriot Daniel Gimeno-Traver was unable to join him in the second round, though, as he went down 6-2 6-4 to Holland's Robin Haase.