We consider a dynamical system described by the differential equation Ẏ_t=−U'(Y_t) with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity ɛ to obtain the stochastic differential equation dX_t^ɛ=−U'(X_t−^ɛ) dt+ɛ dL_t. The process L is a symmetric Lévy process whose jump measure ν has exponentially light tails, ν([u, ∞))∼exp(−u^α), α>0, u→∞. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval (−1, 1). In the small noise limit ɛ→0, the law of the first exit time σ_x, x∈(−1, 1), has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index α=1, namely, ln Eσ∼ɛ^−α for 0<α<1, whereas ln Eσ∼ɛ⁻¹|ln ɛ|^1−1/α for α>1.