Safarova dropped the middle set but went on to outlast Gasparyan 6-3, 4-6, 6-2 indoors at Olympic Stadium in Moscow.
Her next opponent will be fellow Czech Klara Zakopalova, who advanced with a 6-3, 6-3 victory over Romanian Alexandra Cadantu.
Former Moscow champion Francesca Schiavone, however, was among the Day 1 casualties, falling to Poland's Urszula Radwanska 6-3, 6-1.
Monday's other match saw Sweden's Sofia Arvidsson dispatch Serb Bojana Jovanovski 6-2, 6-4.
Beyonce flaunts bikini body, Blue Ivy in vacation pics
Kate Middleton recycles dress at movie premiere