The irreducibility, moderate deviation principle and ψ-uniformly exponential ergodicity with ψ(x) := 1 +||x||₀ are proved for stochastic Burgers equation driven by the α-stable processes for α ∈ (1, 2), where the first two are new for the present model, and the last strengthens the exponential ergodicity under total variational norm derived in Dong et al. (J. Stat. Phys. 154:929–949, 2014).