Conte’s speech during the interval helped his side to bounce back as they re-emerged from the tunnel confident despite the deficit, and scored twice in eight minutes through Brozovic and Vecino.
De Vrij’s wonderful reverse-header from a Candreva corner-kick gave the hosts a first lead of the match with just 20 minutes remaining, and Lukaku’s finish late in stoppage time after a classic counter-attack doubled Inter’s lead – adding insult to Milan’s injury.
The win took Conte’s side level with Juventus in first, while Milan slumped to 10th.
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Milan will be back in action on Thursday, when they host Juventus in the Coppa Italia, before playing Torino on Monday.
While Inter will play Napoli in midweek cup action ahead of their meeting with Lazio in the league on Sunday.