We present a simple criterion, only based on second moment assumptions, for
the convergence of polynomial or Wiener chaos to a Gaussian limit. We exploit
this criterion to obtain new Gaussian asymptotics for the partition functions
of two-dimensional directed polymers in the sub-critical regime, including a
singular product between the partition function and the disorder. These results
can also be applied to the KPZ and Stochastic Heat Equation. As a tool of
independent interest, we derive an explicit chaos expansion which sharply
approximates the logarithm of the partition function.