The Italians have their sights set on qualification for the last 16 and are locked on three points with group leaders Spain before the pair face off in Gelsenkirchen on Thursday.
However, one point from their final two matches should be enough to secure a place in the knockout rounds given that the four best third-place finishers go through. That makes facing Spain - who beat Croatia 3-0 on Saturday - a less daunting prospect for a developing team.
"I'm pleased with them tonight but we shouldn't be too pleased with ourselves," said coach Luciano Spalletti of a solid opening win, albeit one in which they had to come from behind after conceding just 23 seconds into the game.
Italy came into the tournament in Germany as something of an unknown quantity, despite being the title-holders, with a host of fresh faces in the squad.
Spalletti had only a handful of matches to develop his side after replacing Roberto Mancini last year, and the coach's international tournament debut was made even harder by being drawn in a fiendishly difficult Group B.
However, Italy dealt with a hostile crowd in Dortmund which was overwhelmingly Albanian and intent on creating an intimidating atmosphere to rattle their more illustrious opponents.
Huge numbers of Albanians swarmed the streets outside the ground and made an enormous racket inside, easily drowning out Italy's relatively meagre support.
Ahead of the match the head of the Albanian football federation, Armand Duka, said that over half of the Westfalenstadion would be filled with his countrymen but it looked and sounded like much more on Saturday.
And the masses decked out in red and black unleashed a wave of noise when Nedim Bajrami, who plays his club football with Sassuolo in Italy, lashed in the fastest goal in the history of the European Championship to put Albania ahead.
'Kill off games'
Inexperienced Italy could have crumbled after such a rapid setback, and one which was brought upon themselves thanks to Federico Dimarco's sloppy throw-in which allowed Bajrami to rifle past Gianluigi Donnarumma.
Yet instead they fought straight back and were ahead with a quarter of an hour on the clock thanks to a bullet header from Alessandro Bastoni and Nicolo Barella's sumptuous first-time strike.
"It was a bit of a strange start to the match as a goal after 23 seconds could have really hurt the players psychologically," Barella told 'Sky Sport'. "We should have scored more goals and fell away a bit at the end, but we started on the right foot with the win."
The once-feverish Albania support was quickly quietened down and from that point on Sylvinho's team barely threatened at all as Italy calmly took control of the game.
Italy could have won by more as Davide Frattesi clipped a neat finish off the post at the end of a brilliant passing move just after the half-hour mark, while Gianluca Scamacca was denied shortly afterwards by a sharp Thomas Strakosha.
And the one worry for Italy, apart from individual errors, is not capitalising on chances created with two much more accomplished sides awaiting.
Italy were almost dealt a sucker punch in the final moments when Albania substitute Rey Manaj misdirected his lofted finish over Donnarumma after being sent through with a simple long ball over the top. "At this level anything can happen," said Federico Chiesa. "That is where we need to improve, by killing off games that we are dominating."