We investigated the effect of modified aluminosilicates, including bentonite from Armenia (BA) modified with quaternary ammonium salts (BAQAS) and phosphonium salts (BAQPS), on the mechanical properties and morphology of Kevlar/epoxy composites. The Kevlar/epoxy composites containing 1.0 or 3.0 wt.% modified bentonites were fabricated using the hand lay-up technique. The mechanical properties, including the tensile, flexural, and in-plane shear strength, were tested. Based on the obtained results, we found that the mechanical properties increased with modified bentonite loading. The best results were obtained for composites containing 3 wt.% BAQAS, as most of the mechanical properties were significantly improved (tensile strength 302.9 MPa (+30%), Young’s modulus 16.3 GPa (+17%), flexural modulus 23.4 GPa (+12.5%), in-plane shear strength 22.8 MPa (+24.5%), and in-plane shear modulus 677.2 MPa (+42%)). The obtained improvements in the mechanical properties are attributed to the uniform dispersion of the filler, which was confirmed by the highest increase in the intergallery spacing, from 28.3 Å for BAQAS to 45.1 Å for the composite with 3 wt.% BAQAS. Scanning electron microscopy (SEM) analysis of the brittle fracture surface indicated that the addition of modified bentonite to the epoxy matrix changed the morphology of the Kevlar/epoxy/organoclay composites and improved the fiber–matrix interfacial adhesion.