We define the multi-poly-Bernoulli numbers slightly differently from similar numbers given in earlier papers by Bayad, Hamahata, and Masubuchi, and study their basic properties. Our motivation for the new definition is the connection to finite multiple zeta values, which have been studied by Hoffman and Zhao, among others, and are recast in a recent work by Zagier and the second author. We write the finite multiple zeta value in terms of our new multi-poly-Bernoulli numbers.