Roberto Mancini made eight changes from the side beaten by Belgium and was rewarded with a dominant first-half performance, with Kayode and Sebastiano Esposito both making their senior international debuts.
Bastoni opened the scoring after nine minutes, finishing from a corner before Frattesi doubled the advantage in the 22nd minute. Kayode then made his debut even more memorable five minutes later, scoring his first Italy goal to put the visitors 3-0 ahead.
Turkey improved after the interval and Barış Alper Yılmaz pulled one back shortly after the restart, but Italy restored their three-goal advantage almost immediately.
Tonali's effort struck the post and Pio Esposito reacted quickest, heading in the rebound for his sixth goal in 11 appearances.
The result also gave the Esposito brothers a memorable night, with Sebastiano starting alongside Pio for his Italy debut.
"We enjoyed it to the maximum. I tried not to overload him with emotions, we had fun. I'm happy for his debut," Pio Esposito said of playing with his brother.
He also explained the message Mancini had given the team before the match.
"The coach asked us to have fun and do what we felt, and especially in the first half we did that, we combined..." Esposito said.
Esposito also revealed where he would direct the goal he scored after Tonali's shot came back off the post.
"The first one he keeps, then we'll see. I want to dedicate my goal to my grandmother," he said.
Mancini was pleased with the response after the disappointment against Belgium and particularly with the way his players handled difficult conditions in Turkey.
"I am happy because we played really well on a very heavy pitch and it wasn't easy. The boys did some beautiful things, they deserve it," the Italy coach told Sky Sport.
He also praised Kayode's debut, which included the third goal.
"He was very good, it was his debut, it wasn't easy. Very good, beyond the goals," Mancini said.
The Italy coach acknowledged that Turkey's response after half-time made the game more difficult, but his team recovered quickly through Esposito's goal.
"Nothing. This victory is equivalent to a defeat against Belgium. It counts for nothing, the boys understood that they can do good things, this is important," Mancini said when asked what he would take into Friday's meeting with France.
"We have to recover our strength for Friday. It won't be easy."