PHOTO: AFP
Last year's finalists Eintracht Frankfurt also progressed to the last four, as a comedy of defensive errors saw them beat local rivals Mainz 3-0.
Frankfurt and Schalke join Bayern Munich and Bayer Leverkusen in the last four, with the semi-final draw set for Sunday.
Schalke's winner came early in what proved to be a dull game. Just nine minutes in, Austrian Burgstaller broke free on the counter-attack, picking up the ball on the halfway line before keeping his cool in the penalty area to slide the ball into the bottom corner.
Wolfsburg rallied, with Yunus Malli hitting the bar and Ralf Faehrmann forced into two saves before half-time.
The game fizzled out after the break, however, and neither side created another significant chance until late in the second half.
Only in the 85th minute did Wolfsburg come close to equalising, Leon Goretzka clearing Daniel Didavi's header off the goalline.
In Frankfurt, a mistake from Mainz goalkeeper Rene Adler gave the home side the lead on 17 minutes. Making his first appearance since he picked up an injury in October, Adler misjudged a backpass from Stefan Bell, providing Ante Rebic with an open goal.
Eight minutes into the second half, Frankfurt doubled the lead, as Hack turned Sebastien Haller's cross into his own net.
Hack then gave the ball away in his own half just after the hour mark, allowing Omar Mascarell to loft the ball elegantly over Adler for Frankfurt's third.
Mainz's misery was compounded in the 82nd minute, when Danny Latza was sent off for a reckless challenge on Marco Fabian. "We didn't show any balls, and we played like amateurs today," said Mainz midfielder Nigel De Jong. "I am disappointed and frustrated. This is a low point for us."
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