Ag-Ti₃SiC₂ composites are promising electrical contact materials, yet the atomic-scale interfacial behavior between Ti₃SiC₂ and Ag remains poorly understood. Here, first-principles calculations were performed to investigate the interfacial stability, electronic structure, and alloying effects at the Ti₃SiC₂(0001)/Ag(111) interface. Surface-energy calculations for six terminations of Ti₃SiC₂(0001) show that the TiC(TiC) termination is preferred at low carbon chemical potential, whereas the TiC(TiSi) termination becomes the most stable once ΔμC exceeds -1.50 eV. Eighteen interface models combining the six terminations with three stacking sequences (OT, MT, and HCP) were constructed, and their work of adhesion (W_ad) and equilibrium spacing (d₀) were determined by the Universal Binding Energy Relation and full structural relaxation. The HCP stacking is preferred for all terminations, and the C(TiC)-terminated HCP interface is the most stable, with W_ad = 9.25 J/m² at d₀ = 1.2 Å; relaxation enhances W_ad by 10-75%. Charge density, charge density difference, and partial density of states analyses reveal that the interfacial bonding is dominated by C 2p-Ag 4d hybridization accompanied by electron transfer from Ag and Ti atoms to the interfacial C atoms, which accounts for the adhesion hierarchy. Substitutional alloying with Cu, Ni, Zn, and Cr is energetically most favorable within the interfacial Ag layer, and W_ad increases in the order Cu < Zn < Ni < Cr, reaching 11.0 J/m² for interfacial Cr, an enhancement of 19% over the pristine interface. The strengthening correlates directly with the filling of the dopant 3d band. These results provide theoretical guidance for the interfacial design of high-performance Ag-Ti₃SiC₂ electrical contact composites.