Lautaro Martinez scored for the fourth successive game after 22 minutes
before Antonio Candreva punished Dortmund on the break in the closing moments.
Martinez was the best player on view but the Argentina striker blotted his copybook just before Candreva's first Champions League goal when his poor spot-kick was saved by Dortmund goalkeeper Roman Burki.
Antonio Conte's side knew nothing other than victory would do after picking up only one point in their opening two games against Barcelona and Slavia Prague.
Conte was an animated figure on the touchline, and Inter matched his passion as they moved level with Dortmund on four points as Group F reached its halfway stage.
Dortmund recalled Jadon Sancho after the England winger missed Saturday's Bundesliga win over Borussia Monchengladbach for disciplinary reasons.
Dortmund were determined to silence the San Siro crowd in the opening exchanges, and Lucien Favre's side were compact while attempting to utilise the pace of Sancho and Achraf Hakimi down the right.
Romelu Lukaku had little space to run into, and Inter were unable to muster an attack until Candreva's cross almost picked out Martinez.
But Inter were ahead three minutes later when Martinez chested down Stefan De Vrij's pass over the top of the Dortmund defence and slid the ball under the advancing Burki.
Martinez was close to being offside, but his fifth goal in four games stood after a lengthy VAR check.
Quick feet from Sancho set up the visitors' first goal attempt, but Axel Witsel's rising drive cleared the crossbar.
Dortmund's best chance arrived in first-half stoppage time when Hazard slipped in Sancho and Samir Handanovic reacted well to make a smart save low to his right.
Candreva sent a snapshot over at the start of the second half and Inter were largely in control with little threat on their goal.
But Handanovic did well to bat away a deflected Brandt effort that he saw late.
Martinez stepped up to close the game out, but his spot-kick was too close to Burki as the goalkeeper guessed correctly to his right.
Dortmund had hope that Martinez's miss might provide them with a lifeline, but that was quickly snuffed out as Inter launched a lightning counter-raid.
Marcelo Brozovic freed Candreva and the impressive Italian defender made no mistake, galloping on from halfway to bury the ball high past Burki.