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](https://arxiv.org/abs/hep-th/9611043) & [this](https://physics.stackexchange.com/q/21261/2451) Phys.SE post.