Yes. Suppose that $S \subset \partial M$ is the surface we want to attach along. Hatcher's theorem, 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 gives a closely related result (but with stronger hypotheses and stronger conclusions).