The result means Italy, who failed to qualify for both the 2018 and 2022 World Cups either side of their Euro 2020 triumph, avoid another embarrassing tournament miss.
The two nations finish level on 14 points behind England in Group C, but Italy have the better head-to-head record having won 2-1 in Milan in September.
"We knew if we were unable to break the deadlock in the first hour, it would become really tough. Ukraine then made it a more physical match," Frattesi told RAI Sport.
"We were under pressure, but overall it was a good performance. What mattered was that we qualified...
"We suffered, but it was important to get through. Now we'll try to repeat something fantastic."
The final score belied the true nature of the match, with Italy's high-octane attack and Ukraine's counter-attacking threat creating several chances for both teams.
Ukraine had a late penalty appeal turned down, despite Italy midfielder Bryan Cristante looking to have made contact with Mykhailo Mudryk's foot in injury time.
"From my point of view that was a penalty, but again I was not there and it's only about my emotions," Ukraine coach Serhiy Rebrov told a post-match press conference.
Italy travelled to Leverkusen knowing they would progress by avoiding defeat, while Ukraine knew only a win would suffice.
New Italy coach Luciano Spalletti has only been in charge for three months after Roberto Mancini surprisingly left for the Saudi Arabia national team job.
"We had to keep attacking - but Ukraine proved they are a very good team who gave us a lot of problems," Spalletti, who led Napoli to their first Serie A title since 1990 last season, said.
Italy will go into either pot three or four for the tournament draw, which will take place on December 2nd in Hamburg.
Spalletti added his team would "have a moment to pat ourselves on the back" before turning their focus to the finals in Germany, the site of their 2006 World Cup triumph.
"Now, the level will step up. Now, the fun begins."
Midfielder Nicolo Zaniolo said his side's nervy qualification campaign, which included two losses to England and a draw with North Macedonia, would not dampen their hopes of defending the title.
"We had the right motivation. Italy must go to the Euros not just to take part, but to compete as reigning champions."
Ukraine head into play-offs
Ukraine, forced to take the fixture to Germany due to the ongoing conflict with Russia, can still qualify, but will need to navigate the play-offs in March 2024.
Mudryk was lively down the left for Ukraine and crafted a number of half-chances early, with his side denied by poor finishing.
Italy's dominance of possession started to tell after half an hour, Federico Chiesa sliding a smooth pass across the face of goal which eluded a desperate Davide Frattesi.
Italy continued to attack early in the second half but as the game wore on the increasingly nervous Italians sat back, allowing Ukraine more space up front.
Mudryk was always a threat and appeared to have been brought down in the box after contact from Cristante, but the referee waved his team's desperate pleas for a spot-kick away.
