Spalletti was sacked in June following a heavy defeat by Norway in qualifying, with former Milan and Napoli manager Gattuso then leading the Italians to two wins over Estonia and a victory against Israel.
"I trust Gattuso. I'm sure that, as he's already shown, he'll find the right solutions to keep this journey going and achieve what he wants, what he dreams of," Spalletti told reporters on Sunday.
"He's also adjusted a few things tactically in these last matches, because having the forwards closer together has given solutions, especially in games where we faced difficulties - like against Israel - in terms of their playing style.
"They did well, finding more direct plays that allowed us to win."
Italy, who are looking to return to football's showpiece tournament for the first time since 2014, are second in Group I with 12 points from five matches.
They face third-placed Israel in Udine on Tuesday with a win cementing second spot and a place in the playoffs at the very least.
Norway have a six-point lead at the top of the group but have played a game more than Italy.