Jerome Boateng put Germany a goal ahead early on in Lille before Mesut Ozil missed from the penalty spot, but Mario Gomez and Draxler added two more to seal a comfortable last-16 win.
Low made one change to the side which beat Northern Ireland, with Draxler taking the place of Bayern Munich attacker Mario Gotze.
Die Nationalmannschaft were impressive from the off as they pushed for an early goal, with Draxler teasing the Slovakian defence on both flanks.
Slovakian goalkeeper Matus Kozacik had to be alert to keep out Low's men in the opening stages, but Germany were soon ahead from an unlikely source.
Toni Kroos' corner eight minutes in was only partially cleared, and Boateng met the ball first-time to volley in from 25 yards via a deflection.
Germany then had the chance to make it two from the spot after Martin Skrtel had fouled Gomez, but Kozacik guessed right to save a weak spot kick from Ozil - the second Arsenal player to miss at Euro 2016 in two days after Granit Xhaka's shootout woe for Switzerland.
Germany dominated proceedings, but Manuel Neuer had to be alert to make a fine save moments before Gomez all but sealed the win.
Neuer made a commanding stop from Juraj Kucka three minutes before the interval, and Germany responded with a goal in less than 60 seconds as Draxler bamboozled the Slovakian defence to tee up GOmez for a simple tap-in.
Slovakia battled gamely after the break to try and find a way back in, but Draxler got the goal his performance deserved just past the hour-mark.
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Sami Khedira's deflected header looped up in the Slovakian area 63 minutes in, and Draxler reacted quickest to acrobatically volley past a blameless Kozacik.
Germany continued to press for a fourth as they looked to better their European Championship record of a 3-0 win, but Low's side look the team to beat in France given the ease with which they reached the last eight.
