Falla had lost his last five matches on the ATP Tour, including a first-round setback to Stephane Robert at Wimbledon.
He ousted Malisse 6-1, 3-6, 6-3 to set up a second-round match with Matthew Ebden, who won his first-round appearance Monday.
Eight matches were played Tuesday to complete the first round with Ruben Bemelmans earning a chance against top-seeded Janko Tipsarevic by eliminating Nicolas Barrientos 7-5, 6-3.
Qualifier Emilio Gomez won for just the second time this year with a 6-4, 6-4 decision over Eduardo Struvay. Gomez will meet No. 2 seed Kevin Anderson in the second round.
In other matches Vasek Pospisil won over Michal Przysiezny 6-3, 3-6, 6-1, Victor Estrella Burgos was a 7-5, 6-3 winner over Facundo Arguello, Ivo Karlovic defeated Juan Ignacio Londero 7-6 (7-5), 6-4, James Duckworth eliminated Aljaz Bedene 6-3, 6-4, and Chris Guccione advanced when Evgeny Korolev retired in the second set.
Notable deaths of 2014 [PHOTOS]