Gauss law does not completely define electric field: it only defines the conservative component. Similary Faraday's law defines the non-conservative component, that gives 0 when integrated over suface of any volume.
So induced electric field is can't be determined by Gauss law, but does not violate Gauss law, because it contributes 0 to both sides od the equation.
another way to think about it is that if the charge enclosed in the closed surface referred to by Gauss's Law were constant, then we know that Gauss's Law is accurate and there is no net current that crosses the surface boundary. but there can be a current flowing in that is equal to the current flowing out. if that current is not time variant, then Gauss's Law still works, but if the current flowing into the closed surface remains equal to the current flowing out but they're varying in time, then Gauss's Law needs Faraday's Law to fully describe what's going on. really you need all four of Maxwell's Eqs. to fully describe what's going on.
But Gauss's law only gives the divergence... it must be supplemented by an equation giving the curl. So, although all electric fields satisfy Gauss's law, Gauss's law is not enough to determine the fields in all cases.