Tietze extension theorem for Hilbert $C^*$-modules

We prove the following generalization of the noncommutative Tietze extension theorem: if $V$ is a countably generated Hilbert $C^*$-module over a $\sigma$-unital $C^*$-algebra, then the canonical extension $\overline{\Phi}$ of a surjective morphism $\Phi : V \rightarrow W$ of Hilbert $C^*$-modules to extended (multiplier) modules, $\overline{\Phi} : V_d \rightarrow W_d$, is also surjective.