The number eight seed came through 6-2 7-6 6-3 after two hours of tennis.
Del Potro enjoyed the better of the early exchanges, breaking Levine's first two service games to open up a 4-0 lead. Levine clawed two games back on his serve, but Del Potro held to take the first set 6-2.
In the second set Levine improved, breaking Del Potro in the fourth game. At 5-4 ahead and with the opportunity to serve out the set, Levine was broken by the Argentinian before it entered a tie-break that Del Potro claimed 8-6.
Del Potro opened up an advantage early in the third set, breaking his opponent in the second game before moving 5-2 ahead.
Despite Levine breaking in the eighth game, Del Potro responded with a break of his own to wrap up the match and book his place in the next round.