You need geometrisation to prove this fact. See Corollary 12.9.5 [here][1] for a reference. 

You can't prove this without Perelman, at least with our present knowledge. For instance, if the orientable cover is $S^3$, then you must ensure that $M$ be elliptic, and that's precisely the <i>space form conjecture</i>, which is "one third" of geometrisation. But even when the finite cover is some other Seifert space, I don't see an easy argument to conclude without using geometrisation.

  [1]: http://www.dm.unipi.it/~martelli/Geometric_topology.pdf