High-resolution angle-resolved photoemission spectroscopy was used to study the superconducting energy gap and changes in the spectral function across the superconducting transition in the quasi-two-dimensional superconductor 2H-NbSe2. The momentum dependence of the superconducting gap was determined on different Fermi surface sheets. The results indicate Fermi surface sheet-dependent superconductivity in this low-transition temperature multiband system and provide a description consistent with thermodynamic measurements and the anomalous de Haas-van Alphen oscillations observed in the superconducting phase. The present data suggest the importance of Fermi surface sheet-dependent superconductivity in explaining exotic superconductivity in other multiband systems with complex Fermi surface topology, such as the borides and f-electron superconductors.