Summary: We study the diffusion processes \(\{X^\varepsilon_t\}\) in \(\mathbb{R}^d\) \((d\geq 2)\), where \(\{X^\varepsilon_t\}\) satisfy the stochastic differential equations \[ \text{d} X^\varepsilon_t =b(X^\varepsilon_t) \text{d}t+ \tau (X^\varepsilon_t) \circ \text{d}W_t +\sqrt \varepsilon \sigma (X^\varepsilon_t) \text{d} B_t,\;\varepsilon>0. \] \(\{X^\varepsilon_t\}\) are small random perturbations of the degenerate diffusion process \(\{X_t\}\), which satisfies the stochastic differential equation \(\text{d}X_t= b(X_t) \text{d}t +\tau (X_t) \circ\text{d}W_t\). By means of the auxiliary systems, we obtain the Freidlin-Wentzell mean exit time estimates of \(\{X^\varepsilon_t\}\).