Bruno Cesar pulled the Portuguese league runners-up back into the game when he scored from an indirect free kick midway through the second half but Borussia held on, leaving them on seven points and Sporting with three.
The hosts, without coach Jorge Jesus who was suspended, made a bright start but the Bundesliga side struck the first blow.
Aubameyang outmuscled Ruben Semedo as the two chased the ball, left the defender on the ground and dinked the ball over Rui Patricio with his left foot in the ninth minute.
Sporting thought they had levelled in the 39th minute when Sebastian Coates turned the ball in at the far post from a corner but it was controversially disallowed for a push on goalkeeper Roman Buerki.
Dortmund went further ahead two minutes before halftime as Weigl won a loose ball, evaded a tackle and side-footed a shot into the far corner from the edge of the area, the 21-year-old's first goal for the club since he joined them one year ago.
It was harsh on Sporting and the Lions hauled themselves back into the game after Buerki was forced to pick up a poor back pass from Marc Bartra.
William Carvalho touched the ball to substitute Cesar who blasted it through the wall of Dortmund players on the goal line in the 67th minute.
Bas Dost nearly levelled with a header almost immediately, Buerki did well to save a dangerous Ezequiel Schelotto cross and Sporting then failed to find a finishing touch to a Gelson Martins shot as the Germans clung on.
"Today a young and inexperienced team played with such self-confidence during the first half," said Dortmund coach Thomas Tuchel.
"Sporting took big risks after our goals....after scoring to make it 2-1 their fans got louder."
(Writing by Brian Homewood; Editing by Ian Chadband)
