We analyze the avalanche size distribution of the Abelian Manna model on two different fractal lattices with the same dimension dg=ln 3/ln 2, with the aim to probe for scaling behavior and to study the systematic dependence of the critical exponents on the dimension and structure of the lattices. We show that the scaling law D(2−τ)=dw generalizes the corresponding scaling law on regular lattices, in particular hypercubes, where dw=2. Furthermore, we observe that the lattice dimension dg, the fractal dimension of the random walk on the lattice dw, and the critical exponent D form a plane in three-dimensional parameter space, i.e., they obey the linear relationship D=0.632(3)dg+0.98(1)dw−0.49(3).