The Rossoneri have won the competition five times, although they have not tasted success since 2003.
Despite dominating much of the first half, Sinisa Mihajlovic's men did not break the deadlock at the Stadio Luigi Ferraris until the 50th minute. Bacca threaded a pass through for Niang, who took one touch before slotting the ball into the far corner.
The situation then deteriorated even further for hosting Samp in the 64th minute when having just been cautioned, Bosnian defender Ervin Zukanovic was handed a second yellow card for remonstrating with the referee.
Milan finally made sure of the outcome in the third minute of stoppage time. Keisuke Honda found Bacca inside the box and the Colombian went on to toe-poke the ball in.body check tags ::