For the first time full next-to-leading-order electroweak corrections to off-shell vector-boson scattering are presented. The computation features the complete matrix elements, including all nonresonant and off-shell contributions, to the electroweak process pp→μ+νe+νjj and is fully differential. We find surprisingly large corrections, reaching -16% for the fiducial cross section, as an intrinsic feature of the vector-boson-scattering processes. We elucidate the origin of these large electroweak corrections upon using the double-pole approximation and the effective vector-boson approximation along with leading-logarithmic corrections.