Daniele De Rossi's 12th-minute penalty and Ciro Immobile's second-half header secured a deserved, if unimpressive, win in Palermo.
The result sees Italy and Spain still level on points at the top of Group G after the latter beat Israel 4-1.
Sokol Cikalleshi and Odise Roshi failed in the early stages to test Buffon, winning his 168th international cap to go with 612 games for Juventus and 220 for his former club Parma, before Italy took the lead.
Just 10 minutes in, Andrea Belotti was bundled over by Migjen Basha as he attacked Antonio Candreva's free-kick.
Ledian Memushaj was booked for his protests and, after a long delay while two flares thrown into the penalty area were extinguished and Memushaj was removed from the box, De Rossi assuredly sent Thomas Strakosha the wrong way.
Buffon saved from Roshi, while at the other end, Bellotti and De Rossi sent headers off target. Both missed with further attempts and the half ended with Roshi blazing over.
The familiar pattern continued into the second half as De Rossi headed Candreva's corner over the bar.
Candreva twice drew saves from Strakosha, either side of an ambitious Andrea Barzagli effort over the bar, and Belotti missed the target from another corner as the Azzurri struggled to put the game to bed.
Davide Zappacosta and Lorenzo Insigne were similarly wasteful, the latter with a free-kick into the wall, but finally Italy doubled their winning margin when Zappacosta crossed from the right and Immobile headed home.
Albania belatedly pushed for a way back into the game, but Basha and Burim Kukelj shot wide.
De Rossi headed Insigne's corner wide and Cikalleshi also missed the target as Buffon finished with the 426th clean sheet of his distinguished career.
