Comprehensive studies of the influence of brazing gap widths (100, 50, 20 μm) on the structure and mechanical properties of brazed dissimilar joints between Kovar and stainless steel, utilizing Cu−Mn−4.5Co−2.5Fe filler metal during high-temperature vacuum brazing, are presented. X-ray microspectral analysis reveals a two-phase structure in the brazed seams, when Cu−Mn−Co−2.5Fe filler metal is employed. This structure consists of both the α-Cu phase forming the main zone of the brazed seam (with a 100 or 500 μm gap) and discrete grains of the γ-phase (Fe_хMn_yCo_z)Me. During brazing, mutual-diffusion processes occur at the interphase boundary between the liquid filler metal and the solid base metal. The filler metal becomes saturated with constituent components of the base metal—chromium, cobalt, and nickel—thus, influencing the chemical composition and mechanical properties of the brazed seam. Micro-x-ray spectral analysis indicates that reducing the gap from 100 to 20 μm increases the γ-phase (Fe_хMn_yCo_z)Me content in the seam from 13.20% to 90.90%, while simultaneously decreasing the α-Cu phase from 86.83% to 9.10%. These structural changes positively affect the mechanical properties of the brazed dissimilar joints, leading to increased shear strength in Kovar–stainless steel overlap joints. With a gap size of 20 μm, the brazed samples exhibited failure primarily on the stainless steel (600 MPa).