Milan launched straight into an attack from kick-off, with Turkish midfielder Hakan Calhanoglu setting up Leao to slot home.
The previous fastest goal had been Paolo Poggi's for Piacenza against Fiorentina in December 2001 after eight seconds of play.
Portuguese Leao, who usually plays on the left wing, started at centre-forward with star striker Zlatan Ibrahimovic still out with a leg muscle problem and his replacement Ante Rebic also injured.
It was the 21-year-old's third league goal this season and fourth in all competitions.
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