He will next face in-form Austrian, Dominic Thiem, the sixth seed, in a rematch of last year's semi-final.
The second seed, with new coach Andre Agassi absent from the stands, was too good for the Spanish 19th seed.
The Serb, bidding to become the first man in the professional era to win each of the four grand slams twice, found some resistance in the first set with the third game lasting 13 minutes before Djokovic won it.
"After that (first set) I started playing with fewer errors and that was good. That gave me a lot of confidence," Djokovic said in a courtside interview.
He finished Ramos-Vinolas off with a crosscourt forehand winner on his first match point.
"Now it is the quarters in a grand slam and facing Dominic is not easy," the world number two said.
"He is a one of the best players in the world, especially on this surface. Now I have a day to rest and get ready."
