Robin Gosens proved the matchwinner with the only goal of the game midway through the second-half.
Palladino was pleased with the way his team dug in to win on the night.
How much does this victory weigh on the growth path?
"It was a hard-fought and dirty victory. It was a difficult match. I want to congratulate Genoa and its fans because today it was very difficult to play in this stadium with a crowd that cheered on their players. It was a dirty match, they put us in difficulty on the second balls but these are the victories that give you satisfaction. We needed a victory like this, like real men. These are the victories that make us grow and give us great morale and awareness."
Do you look at the standings?
"I don't like looking at them, I think about Torino."
Did you expect to be fourth?
"I didn't think about it. I look at the growth of the team. We must maintain this humility. Today Quarta gave a speech to the team that gave me the shivers. These are important signs of a group that is growing. We must not feel stronger than we are because Serie A is complicated. We must maintain a low profile and this lightness."
What keys did Quarta touch?
"He said very nice things that remain in the locker room and make you understand the strength of this group. It wasn't easy to create this magic that is being created in this group."
Did you expect these answers from De Gea or Gosens?
"I don't expect answers, I expect confirmations. Sottil had an amazing non-possession phase. Quarta had a crazy game. Those who came in, Adli and Mandragora, did well. We brought in a 2006 like Rubino. We don't need any answers from any player."
Did you expect such a goal machine?
"I'm happy because this is a very solid team. Today in Serie A and in today's football you have to defend and attack together. We've reached a balance and we have to continue like this. I like the offensive phase because the team participates, we were a bit sterile and we lacked the final pass. But it was a bit of a dirty game."
Is there a particular focus on young people?
"I come from the youth sector, I coached for two years in Serie A and I have always believed in young people. Fiorentina has a great youth sector, I have brought many young people to retreat and I follow them. When I asked Rubino to join he opened his eyes wide, then he will bring the pastries. Italian football needs this courage."
Beltran?
"I'm really happy for Lucas because he's a guy who suffered a bit at the beginning and I've always said that I believed in him. We used him first as a left attacking midfielder and then as a striker. Lucas is a second striker, putting him in that position freed him up and I'm happy because he's giving a big hand to the team. He can still grow but what Lucas is doing is important. Everyone must feel involved because I make everyone play."