A finitely generated semigroup S is said to be a semigroup of power growth if there exist positive numbers C and d such that \(\gamma (n)\leq Cn^ d\) \((n=1,2,...)\) where \(\gamma\) (n) is the number of elements in S which can be presented by multiplication of not more than n generators. It is known that a finitely generated group G is a group with power growth if and only if G contains a nilpotent subgroup of finite index. In the paper a similar result for finitely generated cancellative semigroups is proved. More exactly, a finitely generated cancellative semigroup S is a semigroup with power growth if and only if S contains a nilpotent subsemigroup of finite index. Using this result the corresponding theorem for invertible automata is proved.