We prove that the zeta function of an irreducible hypersurface quasi-ordinary singularity f equals the zeta function of a plane curve singularity g. If the local coordinates $(x_1,\dots,x_{d+1})$ of f are ‘nice’, then $g=f(x_1,0,\dots,0,x_{d+1})$. Moreover, the Puiseux pairs of g can also be recovered from (any set of) distinguished tuples of f.