The two sides will go head to head once more in the Coppa Italia on Tuesday, with Piatek potentially in line for his full Rossoneri bow.
Head coach Gattuso would not confirm that Piatek will start ahead of Patrick Cutrone, only repeating that the two would not be paired, but he glowingly drew comparisons with former team-mate Tomasson.
"[Piatek] reminds me of Tomasson, he has very specific characteristics," Gattuso told Milan TV. "Jon never relented in training and he gave us a great hand even when he wasn't starting.
"Piatek has just arrived but I like him, he's a curious lad and he's been looking at the photos at Milanello. He likes to observe, discuss and experience the structure that we have in place here. This is fantastic."
The coach was unhappy with the way Milan performed in the final stages of the league encounter with Piatek on the field, however, believing they played into Napoli's hands.
Gattuso explained: "We must start again from the first 50 minutes in the league match, showing the same kind of character.
"In the final 30 minutes, we stopped trying to dribble and became increasingly stretched. This then allowed Napoli to show all of their characteristics. We have must have the desire to play and prove ourselves, dribbling the ball well."