Kolbeinn Sigthorsson's second half winner lifted Iceland to 13 points, five behind Group E leaders Switzerland and one ahead of Slovenia, who beat Cyprus 2-0 away.
Gylfi Sigurdsson had already gone close for the hosts when Albania took the lead after nine minutes. Forward Valdet Rama received the ball on the left before he cut inside his marker and fired a low shot into the far corner.
Birkir Bjarnason levelled for Iceland just five minutes later, however, when he slammed the ball into the roof of the net after Albania's defence failed to diffuse a low cross.
Sigthorsson then made the most of his second chance in front of goal as he put Iceland ahead early in the second half by deftly steering home Birkir Saevarsson's cross from the right.
Both sides created further opportunities but Iceland held on for a victory that saw them move into a World Cup play-off spot.
Iceland host Cyprus on October 11 before travelling to Norway on October 15 to complete their Group E fixtures. Albania will face Switzerland in their next game before finishing up against the Cypriots in Tirana.