The answer is 'yes' by *Britton's lemma* (see [wikipedia][1] and, more generally, Serre's book *Trees* and Scott and Wall's article 'Topological methods in group theory').  

Since $M$ and $U$ are smooth and compact, $\partial U$ has a product neighbourhood $N(\partial I)\cong \partial U\times I$. Cutting along $\partial U$ realises $M$ as a *graph of spaces*, with vertices corresponding to $\pi_0(M-\partial U)$
 and edge spaces corresponding to $\pi_1(\partial U)$.  Van Kampen's theorem now asserts that the graph of spaces structure on $M$ induces a [graph of groups][2] structure on $\pi_1M$.  Note that part of the definition of a graph of groups is that the edge maps should be injective -- this is where your hypothesis that $\pi_1(\partial U)$ injects come in.

Finally, Britton's lemma (or, more precisely, its generalisation to the context of graphs of groups) implies that the natural maps from the vertex groups to the fundamental groups inject, which is the fact you are asking for.

 


  [1]: https://en.wikipedia.org/wiki/HNN_extension#Britton's_Lemma
  [2]: https://en.wikipedia.org/wiki/Graph_of_groups