We propose a novel integrated scheme for generation of Bell states, which allows simultaneous spatial filtering of pump photons. It is achieved through spontaneous parametric down-conversion in the system of nonlinear adiabatically coupled waveguides. We perform detailed analytic study of photon-pair generation in coupled waveguides and reveal the optimal conditions for the generation of each particular Bell state. Furthermore, we simulate the performance of the device under realistic assumptions and show that adiabatic coupling allows us to spatially filter the pump from modal-entangled photon pairs. Finally, we demonstrate that adiabatic couplers open the possibility of maintaining the purity of generated Bell states in a relatively fabrication-fault-tolerant way.