Competing at just their second-ever Euros, the Red and Blacks are regarded as one of the rank outsiders at this summer's tournament in Germany and find themselves in a challenging group alongside holders Italy, Spain and Croatia.
Few would have envisaged Italy slipping up in their opening group fixture, but the Azzurri were left stunned in the first minute of their European title defence following a sharp start to the contest by Albania.
Federico Dimarco's sloppy throw-in, aimed towards teammate Alessandro Bastoni inside the penalty area, was pounced upon by Bajrami, who nipped in front of the Azzurri defender and took a touch before firing a venomous strike into the top corner from a tight angle.
Bajrami scores fastest-ever European Championship goal
Bajrami's strike made the net ripple after just 23 seconds and his goal is the fastest-ever recorded at the European Championship, surpassing the previous record held by Russia's Dmitri Kirichenko, who scored after 67 seconds against Greece in Euro 2004.
Sylvinho relied on Bajrami's attacking threat in Euro 2024 qualifying as he scored a team-high three goals in seven starts to help the Red and Blacks secure top spot in their group.
Albania's supporters when wild as they celebrated Bajrami's opening goal against Italy, but that strike proved to be the only shot on target from the Red and Blacks in the entire match.
Italy responded well to going a goal behind and restored parity just 10 minutes later when Bastoni met an inviting cross from Lorenzo Pellegrini to head home from close range at the back post.
The European champions then competed turnaround in the 16th minute when midfielder Nicolo Barella lashed home a superb first-time strike from the edge of the area.
Sylvinho's side ultimately struggled to gain a foothold as Italy fought back, and goalkeeper Thomas Strakosha was called into action to make a few important saves after the interval to keep Albania in the contest.
The Red and Blacks did improve in the closing stages, though, and they came agonisingly close to scoring a late equaliser, but Italy captain Donnarumma made a vital save to prevent substitute Rey Manaj from lifting his effort into the net.
Luciano Spalletti's side held on to claim all three points and move joint-top of Group B alongside Spain, who secured a statement 3-0 victory over Croatia earlier in the day.
Albania will now turn their attention to their next group match against Croatia at Volksparkstadion in Hamburg next Wednesday, while Italy are next in action against Spain at the Veltins-Arena Gelsenkirchen on Thursday.