Gennaro Gattuso’s side have 34 points from 20 games and sit in the final Champions League berth, overtaking Roma who are one point behind in fifth and Lazio, who are a further point behind in sixth after their 2-1 defeat at Napoli on Sunday.
Gattuso’s side — unsettled by their defeat to Juventus in Saudi Arabia and mounting speculation surrounding Argentine striker Gonzalo Higuain’s loan move to Chelsea — struggled early against Genoa, who are in 14th position following their 10th defeat of the season.
Milan goalkeeper Gianluigi Donnarumma pulled off key saves and new Brazilian signing Lucas Paqueta also impressing as his long range volley rattled the post just before the break.
Donnarumma again came to the rescue in the second half with a double save on Miguel Veloso and Daniel Bessa.
However Milan got into their stride as the game edged into its final period, with Andrea Conti setting up Borini to tap in with 18 minutes left.
Donnarumma then got his hand to a powerful Veloso strike before Patrick Cutrone sent Suso free on the counter attack, the Spaniard shaking off Genoa‘s defenders to shoot home the decisive goal from an angle 10 minutes later.