The British number one, who was being watched by new coach Ivan Lendl for the first time, controlled large parts of the encounter throughout.
He won nine straight games en route to 6-1 6-3 triumph over his Ukrainian opponent.
"I served pretty well again. It got close in the second set and I stayed focused. He was just going for his shots [and he] hit quite a few winners but I didn't let it get to me," said Murray.
"It's good because I could easily have lost second round against [Gilles] Muller but I managed to fight my way through and played three very good matches. I felt like I was moving well right at the end of the week."
Murray will next be in action at the Australian Open later this month.