Simon did not face a break point in the first set, where he broke Gabashvili once, but dropped his serve twice in the second as Gabashvili leveled.
Gabashvili next plays big-serving Australian Nick Kyrgios, who saved both break points he faced in a 6-4, 6-4 win against Canadian Vasek Pospisil.
Also, Sergiy Stakhovsky of Ukraine had 12 aces as he beat Spaniard Feliciano Lopez 6-3, 2-6, 7-6 (5) and Russian Andrey Kuznetsov had 10 as he downed Frenchman Lucas Pouille 7-6 (1), 6-4.
Stakhovsky will face top-seeded Stanislas Wawrinka in the second round.
Later, veteran Julien Benneteau of France was playing German teenager Alexander Zverev.