Having already won the competition with Valencia – in its previous guise as the UEFA Cup – and Chelsea in 2013, Benitez could match Giovanni Trapattoni's record after his Napoli side cruised into the semi-finals thanks to goals from Jose Callejon and Dries Mertens.
Wolfsburg dominated the opening half and could have wiped out their three-goal deficit in the first half but Ivan Perisic and Nicklas Bendtner both failed to take their chances.
Napoli made them pay five minutes after the break as Callejon converted a Gonzalo Higuain pass - albeit with the aid of a deflection.
Daniel Caligiuri raced down the right and delivered a teasing cross into the penalty area for Perisic – one of five changes – who poked goalwards only for Andujar to claw the ball away.
As Napoli failed to find any momentum, Wolfsburg wasted a second chance to give themselves a lifeline after 32 minutes, Bendtner rising above Raul Albiol but heading the wrong side of Andujar's right-hand post.
Wolfsburg's wastefulness in front of goal was punished early in the second half as Callejon controlled a pass from Higuain and fired into the top-right corner, albeit with the aid of a deflection off Ricardo Rodriguez.
Mertens doubled Napoli's lead with 25 minutes remaining as he flicked home after Andujar had saved his first effort from a superb Higuain knock-down.
With Napoli appearing set for a comfortable end to the tie, Wolfsburg finally found their finishing skills as Klose and Perisic netted to bring them level on the night.
Klose rose majestically to head home a Max Arnold corner after 71 minutes, and from the visitors' next attack Josuha Guilavogui crossed for Perisic to nod past Andjuar from close range.
The goals saved some pride for Dieter Hecking's side, but it is Napoli who progress to the last four as they look to lift the trophy for the first time since 1989.
