We present a next-to-leading-order QCD calculation for e+\nu e\mu+\nu\mu jj production in vector-boson fusion, i.e. the scattering of two positively charged W bosons at the LHC. We include the complete set of electroweak leading-order diagrams for the six-particle final state and quantitatively assess the size of the s-channel and interference contributions in VBF kinematics. The calculation uses the complex-mass scheme to describe the W-boson resonances and is implemented into a flexible Monte Carlo generator. Using a dynamical scale based on the transverse momenta of the jets, the QCD corrections stay below about 10% for all considered observables, while the residual scale dependence is at the level of 1%.