Ruben Amorim made six changes to the Milan side that lost against Bologna on Wednesday and was rewarded with the perfect start.
Just four minutes into proceedings, Moreira worked enough space to deliver a pinpoint cross that fellow incomer Christian Pulisic headed home for his fourth Derby della Madonnina goal.
Things got even better for Milan six minutes later, when the returning Adrien Rabiot met Moreira’s low corner with an exquisite dink that spun in off the post.
Yann Bisseck and Carlos Augusto failed to score with headers as Inter looked for a response, which eventually came when Manuel Akanji flicked Hakan Calhanoglu’s corner towards Alessandro Bastoni, who guided the ball past Mike Maignan in the 22nd minute.
Calhanoglu then had a strike of his own tipped over the crossbar before fizzing in a corner that Maignan awkwardly batted away, and another set-piece resulted in Andy Diouf’s low drive being saved. Maignan then superbly tipped Djed Spence’s attempt wide to ensure the Rossoneri remained ahead going into half-time, and there was still time for Moreira to take centre stage and establish a buffer ahead of the break, sending a beautiful free-kick out of Josep Martinez’s reach.
It looked clear that Inter would need their own moment of quality to regain hope of a comeback, and while Calhanoglu supplied it with a wonderfully clipped ball for Augusto, the wing-back made a mess of his attempted header.
Augusto also sent an ambitious attempt harmlessly off-target, before Nicolo Barella was denied by a brilliant Sankhoun Diawara block and Strahinja Pavlovic tackle.
Milan’s main scare late on was a penalty that was given until a VAR check confirmed that Henrikh Mkhitaryan’s cross hit Mario Gila’s midriff rather than his hand.
Alphadjo Cisse had a late long-range strike kept out by Martinez, but the damage had long since been done as Milan won to perfectly respond to Amorim’s first Serie A loss.
The result leaves the Rossoneri just four points behind Inter, who are joint-top for now, but the Nerazzurri’s first league defeat of the season offers Lazio an opportunity to go three points ahead when they face Cagliari on Sunday.
Flashscore Man of the Match: Diego Moreira (AC Milan)