Lautaro Martinez's goal midway through the first half and a late Antonio Candreva strike were enough to give the Nerazzurri their first win over German opposition since March 2011.
The result moves Conte's side into second in Group F, level on four points with Borussia Dortmund but ahead on goal difference, with Barcelona three points clear at the top after their 2-1 win over Slavia Prague.
Inter's positive start yielded a breakthrough 22 minutes in, when Nico Schulz played Martinez onside from Stefan de Vrij's chip over the top, allowing the striker to finish low past Roman Burki.
Lautaro stays hot, fifth goal in four matches 🔥 pic.twitter.com/G34gWMfP29— Bleacher Report Live (@brlive) October 23, 2019
Dortmund began to take control of possession without threatening a swift equaliser, although Samir Handanovic made a good stop low to his right to deny the returning Jadon Sancho just before half-time.
Julian Brandt had been ineffective as the centre-forward in the absence of Paco Alcacer, but he did bring a smart save out of Handanovic after working some space in the Inter box.
The visitors were simply not doing enough in attack, though, and only a timely block from Manuel Akanji stopped Martinez from doubling Inter's lead on the break.
Sancho almost scrambled in an effort, but Marcelo Brozovic blocked and Candreva hacked the ball clear to preserve Inter's slender lead.
Mats Hummels brought down Sebastiano Esposito on the break but Martinez could not beat Burki from the spot.
Candreva made sure of no late drama, however, blasting past Burki after being played clean through by Brozovic.
Inter called game 💯 pic.twitter.com/jfX3ZhENhn— Bleacher Report Live (@brlive) October 23, 2019