Three goals in 16 minutes, including two from Austria international Martin Harnik, sank eighth-placed Werder.
Stuttgart, who have gone nine games without defeat, are five points ahead of Bayer Leverkusen and Hannover. Fifth position at the end of the campaign would earn a spot in next season's Europa League.
"I am usually more reserved but tonight I have to say that it was a really super match we played," Stuttgart coach Bruno Labbadia said.
"We put them under pressure from the start. It is a very satisfactory feeling for a coach."
Leverkusen, in sixth, and seventh-placed Hanover play on Saturday.
Werder took the lead when Markus Rosenberg connected with a Marko Marin cross from the right after 25 minutes.
It did not take long for the hosts to level, Christian Gentner curling an unstoppable 20-metre shot past keeper Tim Wiese in the 37th.
Stuttgart top scorer Harnik headed in from close range on the stroke of halftime before adding his 17th league goal of the season from a Tamas Hajnal cross eight minutes after the restart.
Substitute Cacau sealed the win a minute from time, heading in after being left unmarked in the box.
"That second goal just before the break was unnecessary," said Werder coach Thomas Schaaf.
"It became very difficult after that. Stuttgart were brimming with confidence and we were far too timid."
Leaders Borussia Dortmund, six points clear at the top, travel to bitter rivals Schalke on Saturday while second-placed Bayern Munich need to beat Mainz to keep their slim title hopes alive.