The world number eight won the first three games on Court Number One and went on to record a further break later in the set as he took advantage of the encounter.
It was a much tighter affair in the second set as breaks of serve were traded. However, when serving to stay in the set, Ramos was broken, which allowed Del Potro to go 2-0 up.
That handed the initiative to Del Potro and he won the opening five games in the third set before he wrapped up the win in one hour and 51 minutes.
He will now face Canada's Jesse Levine for a spot in round three.