We prove upper bounds on the order of convergence of lattice-based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness $$\mathring{\mathbf {B}}^s_{p,\theta }$$ B ˚ p , θ s , which also comprise the concept of Sobolev spaces of dominating mixed smoothness $$\mathring{\mathbf {H}}^s_{p}$$ H ˚ p s as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from $$T_N\left( \mathbb {Z}^d\right) \cap [0,1)^d$$ T N Z d ∩ [ 0 , 1 ) d , where $$T_N$$ T N is a real invertible generator matrix of size d. For such rules, the worst-case error can be bounded in terms of the Zaremba index of the lattice $$\mathbb {X}_N=T_N\left( \mathbb {Z}^d\right) $$ X N = T N Z d . We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to $$\log N$$ log N -factors for arbitrary smoothness s. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.