1960 Albanian Cup (Albanian: Kupa e Shqipërisë) was the eleventh season of Albania's annual cup competition. It began on Spring 1960 with the First Round and ended on May 1960 with the Final matches. KF Partizani were the defending champions, having won their fourth Albanian Cup last season.
The rounds were played in a two-legged format similar to those of European competitions. If the aggregated score was tied after both games, the team with the higher number of away goals advanced. If the number of away goals was equal in both games, the match was decided by extra time and a penalty shootout, if necessary.