Paraguay forward Lucas Barrios and Turkey midfielder Nuri Sahin struck in the second half to leave Bayern 12th in the table, 13 points behind leaders Mainz 05 and facing a tough battle to retain their title.
Bayern, who sorely missed injured forwards Arjen Robben and Franck Ribery, have the worst attacking record in the league with only five goals scored.
"We created the better chances in the first half and we have to keep on fighting," Bayern captain Mark van Bommel told reporters.
Dortmund's sixth win in seven league games kept them second with 18 points, three adrift of leaders Mainz 05.
Bayern coach Louis van Gaal had hoped last week's 2-1 Champions League win at FC Basel could prove a turning point for his side.
Gomez, the Bundesliga's most expensive signing when he joined last season, missed a chance when his first touch let him down as he was clean through on goalkeeper Roman Weidenfeller.
Stuttgart's woes continued when Greek forward Theofanis Gekas headed in a cross by Brazilian Chris to put Eintracht ahead in the 18th minute with his fifth league goal of the season. Chris scored the second with another header after a corner in the 69th minute.
