Silva had been a target for many of Europe's top sides before he joined Milan from Porto in the close-season, but he had failed to score on 16 previous top-flight appearances.
Gattuso believes the weight of history may be behind Silva's slow start to life in Italy, with the likes of Filippo Inzaghi, George Weah and Marco van Basten having adorned the number nine jersey.
"I'm happy that Andre Silva scored his first Serie A goal," the Milan coach told Mediaset Premium .
"It's already hard to carry this prestigious jersey, even more so when you are young, but we must continue our work and grow over time, believing in ourselves."
What a way to take home a win and score your debut goal in #SerieATIM ! ️ See all the pics of the game: https://t.co/htApNwvD08— AC Milan (@acmilan) March 11, 2018
Esiste modo migliore di vincere e segnare il tuo primo gol in A?
Rivivi #GenoaMilan con le migliori della gara: https://t.co/XZeokBss4Y pic.twitter.com/gfFqrA5blu
That defeat was Milan's first in any competition since December, though, and Gattuso confessed he was intrigued to see how his in-form side responded to adversity.
"We were worried and I was curious frankly to see how we'd react to our first defeat in two and a half months. I got a good response," he added.
"I heard a lot of talk about our physical shape against Arsenal, but we simply got the wrong attitude.
"We had prepared a different approach, but got the first half completely wrong and left them far too many spaces. Arsenal have the quality to put the ball where they want if you give them time and space.
"We cannot give gifts to anyone, the team has understood that and we go forward with those lessons. Don't forget we played well, but could easily have lost this game.
"We are not going to London for a walk in the park, we are representing a glorious jersey and want to see what happens next."