The 19-year-old Brit, who has played just two competitive matches since Wimbledon because of a wrist injury which was taped up on Monday, was tested throughout the first set.
Spaniard Dominguez Lino served for the first set at 5-4 but, courtesy of four brutal returns, Robson was able to break her opponent before going on to take the opening set.
That late break appeared to have crushed Dominguez Lino's spirit, with the Spaniard struggling to keep up with Robson's powerful display of tennis throughout the second set.
With victory in her sights, the Brit raced through the second set, handing her opponent in the process.
"I'm really happy with the way that I played today," Robson told BBC Radio 5 live. "Especially in the second set, I really felt like I got a rhythm in the rallies, played very patiently, but then took my chance when the opportunity arose.
"I always knew going into the match that I'd be a little bit rusty because I haven't played in a few weeks.
"Practice points are very different to playing in the first round of a Slam. I just had to stay focused out there and confident in my game-plan."
Robson will now face Caroline Garcia, who is ranked 75th in the world, after the Frenchwoman beat American Shelby Rogers 6-3 6-2.