The important point here is that $E\otimes M_\chi$ has a Verma filtration where $M_{\chi+\nu}$ appears with the multiplicity of the weight $\nu$ in $E$.  The highest weight is at the bottom of the filtration, the lowest at the top. 

Consider the case when $\lambda$ is dominant; this follows by the usual argument that translation functors give equivalences between blocks of category $\mathcal{O}$; Let $E$ be a f.d. representation with $\lambda-\chi$ extremal.  Thus, $M_\lambda$ appears in the filtration on $E\otimes M_\chi$, and no other Vermas in the same block appear, so it's a summand and thus a quotient.

Thus, it suffices to replace $\chi$ with an arbitrary dominant weight; in particular, we can assume that $\chi-\lambda$ is dominant.  Now choose $E$ to be the representation with lowest weight $\lambda-\chi$.  Then $M_\lambda$ is the quotient of $E\otimes M_\chi$ by the submodule generated by all vectors of weight $>\lambda$.  Thus we're done.