The Poland international scored a hat-trick as Jurgen Klopp's side celebrated their coach's commitment to a new two-year contract extension in emphatic style.
Karim Haggui gave the guests the lead in the 12th minute, but it was one-way traffic from then on with Sokratis Papastathopoulos levelling soon after and Marco Reus then giving Dortmund the lead.
Lewandowski scored his three goals before Pierre-Emerick Aubameyang completed the scoring, and a miserable night for Stuttgart, who lost for the first time under their coach Thomas Schneider.
Stuttgart have made a habit of scoring from set pieces and they took the lead when Haggui met Alexandru Maxim's corner to head in from eight yards.
An almost carbon copy saw Dortmund level seven minutes later with Papastathopoulos directing the ball past Sven Ulreich from the same distance.
Just three minutes later, Reus turned on the turbo before sending a low left-footed shot across Ulreich and into the far corner to give Dortmund a deserved lead.
Their lead became more comfortable when Lewandowski scored his first of the night in the 54th minute - adjusting his feet after a cute Reus flick - adding a second from Henrikh Mkhitaryan's perfect pass almost straight from the restart of play to give Dortmund a commanding lead.
A touch of fortune allowed the Pole to complete his hat-trick after Jakub Blaszczykowski's shot fell kindly to him after rebounding off the post, but he was not complaining.
Lewandowski almost claimed a fourth when he got his head to the ball in the 81st minute, but the ball had already crossed the line from Aubameyang's effort as Dortmund hit Stuttgart for six.