Both the acceleration of cosmic rays (CRs) in supernova remnant shocks and their subsequent propagation through the random magnetic field of the Galaxy are deemed to result in an almost isotropic CR spectrum. However, the MILAGRO TeV observatory discovered sharp (~10°) arrival anisotropies of CR nuclei. We suggest a mechanism for producing a weak and narrow CR beam which operates en route to the observer. The key assumption is that CRs are scattered by a strongly anisotropic Alfvén wave spectrum formed by the turbulent cascade across the local field direction. The strongest pitch-angle scattering occurs for particles moving almost precisely along the field line. Partly because this direction is also the direction of the minimum of the large-scale CR angular distribution, the enhanced scattering results in a weak but narrow particle excess. The width, the fractional excess, and the maximum momentum of the beam are calculated from a systematic transport theory depending on a single scale l which can be associated with the longest Alfvén wave, which efficiently scatters the beam. The best match to all three characteristics of the beam is achieved at l ~ 1 pc. The distance to a possible source of the beam is estimated to be within a few 100 pc. Possible approaches to the determination of the scale l from the characteristics of the source are discussed. Alternative scenarios of drawing the beam from the galactic CR background are considered. Finally, the beam-related large-scale anisotropic CR component is found to be energy independent, which is also consistent with the observations.