We show that a randomly perturbed digraph, where we start with a dense digraph $D_α$ and add a small number of random edges to it, will typically contain a fixed orientation of a bounded degree spanning tree. This answers a question posed by Araujo, Balogh, Krueger, Piga and Treglown and generalizes the corresponding result for randomly perturbed graphs by Krivelevich, Kwan and Sudakov. More specifically, we prove that there exists a constant $c = c(α, Δ)$ such that if $T$ is an oriented tree with maximum degree $Δ$ and $D_α$ is an $n$-vertex digraph with minimum semidegree $αn$, then the graph obtained by adding $cn$ uniformly random edges to $D_α$ will contain $T$ with high probability.