Draxler, who joined from Schalke 04 at the start of the season, struck in the 44th minute after a classy dribble took him past midfielder Thomas Foket before he exchanged passes with Vierinha and tucked the ball beyond keeper Matz Sels.
Gent began the second half much improved, pushing for an equaliser, but Draxler netted again in the 54th minute as he capitalised on one of many errant passes by the home defence.
Max Kruse made it 3-0 with a close-range volley and came close to adding a fourth goal when his shot rebounded off the post.
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Sven Kums sparked a late rally by Gent after capping a fine run in the 80th minute by delivering a right-foot drive that beat Koen Casteels at the far post.
Substitute Kalifa Coulibaly then gave Gent hope for next month's second leg when he converted a left-wing cross with a stooping near-post header in the 89th minute.
"We lost the plot and shifted back a gear at the end," Draxler told German state broadcaster ZDF.
"We messed up what would have been an even better result so we have mixed feelings. Still ... we have everything in our own hands."
Gent, making their first appearance at this stage of the competition, appeared nervous at times.
The Belgian top-flight champions defended sloppily from the outset and only turned up the heat on Wolfsburg's inexperienced keeper Casteels towards the end of the match.
"You saw that once we put the pressure on that anything is possible but we failed to do that at the start," said Foket.
"We now know for the second leg that we have to do so from the beginning."
The Germans, however, will fancy their chances of going through when the teams meet in the return game on March 8.
(The story was refiled to correct the headline to 'quarters' from 'last 16')
(Reporting by Robert-Jan Bartunek; Editing by Tony Jimenez)
