The Feynman $i\varepsilon$-prescription is just the first infinitesimal angle of a $e^{i\pi/2}$ Wick rotation in the complex $t$-plane. Heuristically, on physical grounds, no poles are expected after the first infinitesimal rotation, so this in turn is equivalent to the full $e^{i\pi/2}$ Wick rotation.
NB: Wick rotation of spinors are subtle, cf. e.g. arXiv:hep-th/9611043 & this Phys.SE post.