quantum field theory - How can perturbativity survive renormalization?

The most usual way to renormalize quantum field theories is by re-writing the Lagrangian in terms of physical (finite) parameters plus counter-terms. Take $\lambda \phi^4$ theory for instance:

$$ {\cal L} = {1\over2}(\partial_{\mu}\phi)^2-{m^2 \over 2} \phi^2 - {\lambda\over 4!}\phi^4 + {\cal L}_{CT}, $$

$$ {\cal L}_{CT} = {\delta Z\over2}(\partial_{\mu}\phi)^2-{\delta m \over 2} \phi^2 - {\delta\lambda\over 4!}\phi^4. $$

All parameters with $\delta$ in ${\cal L}_{CT}$ are divergent quantities. Then what we do is to treat everything in ${\cal L}_{CT}$ as interactions and calculate it perturbatively.

My question is: how can we do that? Considering that the "couplings" in this case ($\delta Z$, $\delta m$ and $\delta \lambda$) are huge numbers?