The striker completed a move to San Siro on Wednesday, agreeing a four-and-a-half-year deal after Milan paid Genoa a fee believed to be in the region of €35 million (£30m/$40m).
Piatek's transfer went through on the same day that Gonzalo Higuain cut short his loan move from Juventus to reunite with Maurizio Sarri at Chelsea, and the Poland international joins the Rossoneri having netted 13 goals in 19 Serie A games this season.
And the 23-year-old already has his sights set on European club football's premier competition, which has not been graced by seven-time champions Milan since 2013-14.
Gennaro Gattuso's side occupy the fourth and final qualification berth, with a one-point advantage over fifth-placed Roma.
"I was born ready," Piatek told a news conference on Thursday. "I will do everything to return AC Milan to the Champions League.
"I didn't think I'd ever get here. Next to me is [Paolo] Maldini, whom I've always admired. For me it's a great honour to be here. I hope I can show my talent in every game...but everything will be decided on the field.
"It's a pleasure for me to be here. I'm a striker, I want to score in every game, work very hard and then we'll see what happens on the field.
"I've always believed in myself. I wanted to show something good in Serie A when I joined Genoa. It's great to play at San Siro, but I think everything will always be about what happens on the pitch. I'm focused on that and I'll do everything to do my best and score goals."
Piatek confirmed that pictures of him as a boy with a Milan towel were genuine, while he revealed his surprise at attracting attention from San Siro after only half a season in Italy.
"The picture of me wearing an AC Milan towel was real. It was always my dream to be here," he said.
"This is football for you, unexpected. I never thought after scoring 13 goals other clubs, such as Milan, would be interested in me but this is the beauty of football."