No, the quantum action is not invariant. The non-invariance of the path-integral measure introduces a symmetry-breaking term.
The perturbative leading-order treatment makes the whole issue clearer. Take QED and a triangle graph with one axial-vector current and two vector currents. That's related to the pion via the PCAC hypothesis, which states
$$\partial_{\mu} j_5^{a,\mu}=f_{\pi} M_{\pi}^2 \pi^a, \quad j_{5,\mu}^a=\overline{\psi} \gamma_{\mu} \gamma_5 \frac{\tau^a}{2} \psi.$$
Here ##f_{\pi} \simeq 92 \; \mathrm{MeV}## is the pion-decay constant, ##\pi^a## is the pion field and ##j_{5,\mu}^{a}## the axial-vector current.
Plugging this into the triangle graph, you find that it's linearly divergent, and that implies that one has to regularize the diagram. Now the Ward-Takahashi identity for the vector current, which after all is coupling to the electromagnetic field must hold. So we have to regularize the diagram in a way to preserve the electromagnetic gauge symmetry. One unambigous way is Pauli-Villars regularization, i.e., you subtract the diagram with large fermion masses. This violates obviously the chiral symmetry for vanishing fermion masses but keeps the vector currents conserved. This is the unique right choice of regularization since the vector current's conservation is a necessary condition to keep the local gauge invariance, which must not be broken, because otherwise the model would break down.
This adds an additional term to the PCAC relation which, written in terms of the corresponding external fields (two photons coupled to the vector currents and a pion via the PCAC relation to the axial-vector current), reads
$$\partial_{\mu} j_5^{a,\mu} = f_{\pi} m_{\pi}^2 \pi^a - \alpha \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma} \mathrm{Tr}(Q^2 \tau^a/2).$$
The ##Q## are the charges of the fermions ("quarks").
In this case the anomaly is a great feature, because if there wouldn't be this anomalous term, the ##\pi_0 \rightarrow \gamma \gamma##-decay rate would come out way too low, which would imply that the PCAC hypothesis, working great otherwise due to (approximate) chiral symmetry of the light-quark sector of QCD, may be wrong, but the anomaly saves the day. Using the result of the triangle calculation, leads to a decay width
$$\Gamma(\pi^0 \rightarrow \gamma \gamma)=\frac{\alpha^2}{128 \pi^3} \frac{m_{\pi}^3}{f_{\pi}^2} \frac{N_{\text{color}}^2}{9}.$$
Putting the measured values of the parameters, i.e., ##f_{\pi}=92 \; \text{MeV}##, ##m_{\pi}=135 \; \text{MeV}##, ##N_{\text{color}}=3##, leads to ##\Gamma(\pi^0 \rightarrow \gamma \gamma) \simeq 7.81 \; \mathrm{eV}##, which compares well with the measured value. This shows that everything is consistent with this decay and the strong-interaction physics, including the PCAC hypothesis (i.e., the assumption of chiral spontaneously broken symmetry) and the number of colors being 3 in the standard model QCD.
