The structural and magnetic properties of disordered La2CuMnO6 double perovskite oxide are reported. Temperature variation of dc magnetization in zero field cooled and field cooled protocols manifest the cluster glass feature of the compound. The cooperative glassy magnetic state is explored by the memory effect in thermal variation as well as relaxation dynamics. The frequency dependent shift of the ac susceptibility peak satisfies dynamical scaling law with Tc ~ 26.4 K and dynamic exponent zν ~ 3.26 proposing a cluster-glass like state. The anti-site disorder and competing exchange interactions of two different transition metal have been argued for the observed cluster glass state in this B-site disordered La2CuMnO6 double perovskite.