We consider the Navier-Stokes equations in the 3D torus perturbed by a super-linear noise in Stratonovich form. We show global-in-time existence and uniqueness of smooth solutions, ruling out the possibility of finite-time blow-ups for such equations. The result is an application of the abstract framework developed in (Bagnara A suitable nonlinear Stratonovich noise prevents blow-up in the Euler equations and other SPDEs. Preprint, 2023. arXiv:2312.10446), where it is shown that a suitable Stratonovich noise prevents the blow-up possibly induced by a drift with super-linear growth in a class of SPDEs.