  Properties of the noise-driven escape kinetics are mainly determined by the
stochastic component of the system dynamics. Nevertheless, the escape dynamics
is also sensitive to deterministic forces. Here, we are exploring properties of
the overdamped drifted escape from finite intervals under the action of
symmetric $\alpha$-stable noises. We show that the properly rescaled mean first
passage time follows the universal pattern as a function of the generalized
P\'ecklet number, which can be used to efficiently discriminate between domains
where drift or random force dominate. Stochastic driving of the $\alpha$-stable
type is capable of diminishing the significance of the drift in the regime when
the drift prevails.
