We study oriented percolation on random causal triangulations, those are random planar graphs obtained roughly speaking by adding horizontal connections between vertices of an infinite tree. When the underlying tree is a geometric Galton--Watson tree with mean $m>1$, we prove that the oriented percolation undergoes a phase transition at $p_c(m)$, where $p_c(m) = \fracη{1+η}$ with $η= \frac{1}{m+1} \sum_{n \geq 0} \frac{m-1}{m^{n+1}-1}$. We establish that strictly above the threshold $p_c(m)$, infinitely many infinite components coexist in the map. This is a typical percolation result for graphs with a hyperbolic flavour. We also demonstrate that large critical oriented percolation clusters converge after rescaling towards the Brownian continuum random tree. The proof is based on a Markovian exploration method, similar in spirit to the peeling process of random planar maps.