Yes.  Suppose that $S \subset \partial M$ is the surface we want to attach along.  [Hatcher's theorem][1], correctly generalised, says that the set of "boundary slopes" in $S$ (curves which are part of an "essential" surface in $M$ with boundary only meeting $S$) is a "thin" subset of all curves in $S$.  So we can attach a two-handle to $S$ whose core is not equal to any of these boundary slopes.  After attachment, the new manifold is again irreducible.  Induct.

Lackenby's paper [Attaching handlebodies to 3-manifolds][2] gives a closely related result (but with stronger hypotheses and stronger conclusions).


  [1]: https://pi.math.cornell.edu/~hatcher/Papers/bdyslopes.pdf
  [2]: https://arxiv.org/abs/math/0109059