- Davide Frattesi (12', 29')
- Andriy Yarmolenko (41')
Italy are now second in the standings on seven points ahead of Ukraine on goal difference and with a game in hand on all their group rivals. England top the section with 13 points.
"There was a lot of pressure in recent weeks, and I had to understand many things in a short time," Spalletti told RAI.
He said he was pleased with the performance despite the narrow win, adding: "We have to be happy because it's not always possible to not even concede a single cross to our opponent."
Italy broke the deadlock in the 12th minute through Frattesi after Mattia Zaccagni retrieved the ball on the left and rolled it back to the midfielder, who scored from inside the box.
Just before the half-hour mark Frattesi got his second goal by calmly slotting home once again from close range, capitalising on a ricocheted shot.
Frattesi, who has only appeared for his new club Inter Milan as a substitute so far this season after joining from Sassuolo in July, downplayed his contribution.
"I still have a lot to improve. It's normal not to be a starter at Inter as I've just joined them, but these games will help me grow a lot," he said.
Ukraine pulled one back four minutes before halftime when Andriy Yarmolenko put a rebound into an empty net from close range to increase the tension among the home fans.
The visitors exerted considerable pressure after the restart in their quest for an equaliser but only managed to create half-chances that Italy managed to deal with.
Manuel Locatelli came close for Italy towards the end of the match with a powerful attempt that struck the crossbar following several earlier efforts that were blocked inside the box.
"In the final minutes, they [Ukraine] left several players in attack as losing with a larger goal difference wouldn't change much for them. However, we need to be better at scoring the third and fourth goal when we can," Spalletti said.
- Alejandro José Hernández Hernández