Fourth-seeded Rublev won after he was pushed hard by Giron as the first set went with serve, before breaking the American twice in the second set. It was Rublev's first match since he was beaten by fellow Russian Daniil Medvedev in the Australian Open quarterfinal.
Rublev will play Andy Murray in the second round.
Khachanov's win sets up another Russian-British match in the second round, this time against qualifier Cameron Norrie.
Alex de Minaur won an all-Australian match against John Millman 6-1, 6-4 to set up a second-round meeting with Kei Nishikori. Poland's Hubert Hurkacz saw off experienced French player Adrian Mannarino 6-3, 7-6 (6).