This follows from Noam's answer and the classical zero-free region for Dirichlet $L$-functions. For a specific reference, see Davenport's *Multiplicative Number Theory*, Chapter 14. The result quoted below can be found on page 93:

**Theorem.** There exists a positive absolute constant $c$ with the following property. If $\chi$ is a complex character modulo $q$, then $L(s,\chi)$ has no zero in the region defined by
$$
\sigma \geq \begin{cases}
1 - \dfrac{c}{\log(q |t|)} & \text{if $|t| \geq 1$},\\[1em]
1- \dfrac{c}{\log(q)} & \text{if $|t| \leq 1$}.
\end{cases}
$$
Note that this holds for complex characters because in this case there is no exceptional zero. A similar result holds for quadratic characters, aside from the exceptional zero, of course.

We thus have, for any non-exceptional zero $\rho = \beta+i\gamma$
$$
|1-\rho| = |1-\beta - i\gamma| \geq 1-\beta \geq \frac{c}{\log(q)}
$$
for $|\gamma|\leq 1$.