The effect of pressure on the phase behavior of a multicomponent polymer blend was studied by small-angle neutron scattering (SANS). The blend was composed of saturated polybutadiene with 89% 1,2-addition (component A), polyisobutylene (component B), and an A-C diblock copolymer, where block A was chemically identical to component A and block C was a saturated polybutadiene with 63% 1,2-addition (sPB63). At atmospheric pressure, the blend forms a lamellar phase at low temperature, forms a microemulsion phase at intermediate temperatures and is macrophase separated at high temperatures. No evidence of homogeneous phases was found at atmospheric pressure. Upon pressurization the A/B/A-C blend exhibited a homogeneous phase across a wide range of pressures and temperatures. The pressure dependencies of the Flory-Huggins interaction parameters in this system (chi(AB), chi(AC), chi(BC)) were determined from SANS measurements on binary blends and used to model the thermodynamic properties of the multicomponent blend as a function of temperature and pressure with the random phase approximation, self-consistent-field theory, and Flory-Huggins theory. We demonstrate excellent agreement between theory and experiment without any adjustable parameters.