Summary: In this paper, we consider the non-linear Chern-Simons-Schrödinger equation \[ -\Delta u+\left(\frac{h_u^2(|x|)}{|x|^2}+\int^{\infty }_{|x|} \frac{h_u(s)}{s}u^2(s){\text{d}}s\right) u=-a|u|^{p-2}u+f(u),\,\, x\in \mathbb{R}^2, \] where \(a>0\), \(p\in (2,3)\) and \[ h_u(s)=\int^s_0\frac{\tau }{2}u^2(\tau )d\tau =\frac{1}{4\pi }\int_{B_s}u^2(x){\text{d}}x \] is the so-called Chern-Simons term, \(f: \mathbb{R}\rightarrow \mathbb{R}\) has critical exponential growth which behaves like \(e^{\alpha u^2}\). We establish a new version of Trudinger-Moser inequality in the working space \[E:=\left\{ u: u(x)=u(|x|), \int_{\mathbb{R}^2}|\nabla u|^2\text{d}x<\infty , \int_{\mathbb{R}^2}|u|^p\text{d}x<\infty \right\}\]of the associated with the energy functional related to the above equation, and prove that the energy functional is well-defined in \(E\). By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a mountain-pass type solution for the above equation under some weak assumptions.