Jurgen Klopp's side were given an early scare when Karim Haggui gave the visitors a 13th-minute lead, but two goals in three minutes from Sokratis Papastathopoulos and Marco Reus turned the clash around by the interval.
Lewandowski netted a quickfire double of his own after the break to put Dortmund in complete control, and he added a third to put the gloss on a victory which maintains his side's 100% home record.
A set piece at the other end provided last season's Champions League finalists with their equaliser, as Sokratis stooped to head Sahin's delivery low past visiting goalkeeper Sven Ulreich.
The comeback was complete nine minutes after Dortmund had gone behind when Reus drilled a shot into the bottom corner after he had fortuitously dribbled through Haggui and Christian Genter.
When play resumed after the interval, it was the Lewandowski show, which began with the Polish international striker scoring twice in the space of 90 seconds.
His first came following a one-two with Reus, whose return back-heel was converted by Lewandowski, who soon capitalised on some poor Stuttgart defending to get on the end of Henrikh Mkhitaryan's through-ball and fire another past Ulreich to make it 4-1.
An 18-minute treble was completed by the prolific striker when he tapped in the rebound of Jakub Blaszczykowski's shot on 72 minutes, and Pierre-Emerick Aubameyang put the final nail in the Stuttgart coffin by shooting, via a deflection off Moritz Leitner, over Ulreich on 81 minutes.
Dortmund are now two points clear of rivals Bayern Munich at the top of the German league table.