Davide Frattesi almost extended the Italians' lead in the 34th minute but Albanian goalkeeper Thomas Strakosha tipped his chipped shot on to the post.
Despite fielding a team packed with Serie A players, Albania were left chasing shadows for much of the game as the new-look side assembled by Italy coach Luciano Spalletti dominated with neat and fast passing, albeit without creating many chances.
Central defender Riccardo Calafiori impressed on only his third international appearance as he took the place of more experienced team mates out with injuries.
Spalletti, who led Napoli to the Serie A title a year ago, took over the national team in September when Roberto Mancini controversially switched to coach Saudi Arabia, having guided Italy to victory at Euro 2020.
Spalletti praised his players for not allowing the early setback to upset them.
"They were great because they all shared in the difficulty that the team was facing and they did the right thing. No one threw their arms up in air," he said.
"They showed that they are a strong team in terms of character. I am happy with them. But there are lots of things that need to improve."
Albania coach Sylvinho said his players had felt the tension as the national team embarked on only their second major tournament.
"It wasn't easy for us," - the former Brazil told Sky Sport television. "We have young players and there was some nervousness. Italy are a strong side and we were in a battle with them. The lads did well."
The tournament in Germany is an important stepping stone for Italy's hopes of returning to the World Cup after failing to qualify for world soccer's showpiece event in 2018 and 2022.
Victory for Italy was vital in the toughest group in the tournament and a bigger test will come on Thursday when they face Spain, who comfortably beat Croatia 3-0 in Berlin. Albania play Croatia on Wednesday in Group B.
Published 15 June 2024, 23:23 IST