Gattuso's sister Francesca died aged 37 earlier this month from what was only described as an "illness" and she had spent four months in intensive.
The Napoli coach dedicated his side's progression to her, while hailing his players.
Dries Mertens became the Serie A club's outright all-time leading goalscorer, cancelling out Christian Eriksen's opener in Sunday's (AEST) 1-1 draw at Stadio San Paolo.
The result was enough to send Napoli through to the final against Juventus thanks to a 2-1 aggregate win in the semi-final."I dedicate this result to my sister and my family," Gattuso said.
"The guys were extraordinary for the dedication they put on the pitch. I thank them because they played for me and they were close.
"I have a team that has great quality, but there was [a will] to suffer because not all the mechanisms went in the right direction. After the first very naive goal conceded it was not easy to go back up.
"We showed character and I have to say that in terms of intensity we started from where we left off. We went through a very hard period, it wasn't even easy to train for the health emergency."
Mertens's 122nd goal for Napoli saw him pass Marek Hamsik (121) as the club's outright all-time leading goalscorer.
Gattuso was pleased with the way his side performed after the Belgian's 41st-minute strike.
"After Mertens' goal we had a sort of liberation and we expressed ourselves better," he said.
"The boys are giving everything both in terms of commitment and availability. Now we will play the final with our weapons. If we continue on this path we can take great satisfaction from ourselves."
Napoli will face Juventus in the final in Rome on Thursday (AEST).