The Poland striker, who is top of the scorers' list with nine goals, scored three times in just 18 minutes to lift Dortmund to 28 points, two ahead of Bayern Munich who are in action on Saturday.
Stuttgart, who suffered their first defeat in eight league games under coach Thomas Schneider, took an unexpected early lead when Karim Haggui headed in an Alexandru Maxim corner but any joy was shortlived.
Dortmund, who host Arsenal in the Champions League next week, reacted almost instantly, scoring twice in three minutes to go in front leaving the Stuttgart defence in tatters with their pace.
Defender Sokratis Papastathopoulos levelled with a well-taken header in the 19th minute before Marco Reus slotted in his sixth league goal after a solo run.
They picked up where they left off nine minutes after the restart with Reus turning provider to flick the ball through to Robert Lewandowski to slot in.
The Poland striker added another two minutes later when he chased down a through ball from Henrikh Mkhitaryan to beat keeper Sven Ulreich.
He completed his hat-trick in the 72nd after Jakub Blaszczykowski's shot was palmed onto the bar by Ulreich and Lewandowski beat Stuttgart's defenders to the ball to tap in.
Substitute Pierre-Emerick Aubameyang got his seventh goal of the campaign when he lobbed the ball over Ulreich in the 81st minute as Dortmund matched their own club record of six straight home wins since the start of the season.
Treble winners Bayern travel to Hoffenheim where they will look to stay unbeaten for a 36th straight game and equal a 30-year-old Bundesliga record set by Hamburg SV in 1982/83.
Third-placed Bayer Leverkusen take on bottom-placed Eintracht Braunschweig.
(Reporting by Karolos Grohmann; editing by Toby Davis)
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