A methodology to investigate the linear viscoelastic properties of complex fluids at elevated pressures (up to 120 MPa) is presented. It is based on a dynamic light scattering (DLS) setup coupled with a stainless steel chamber, where the test sample is pressurized by means of an inert gas. The viscoelastic spectra are extracted through passive microrheology. We discuss an application to hydrogen-bonding motif 2,4-bis(2-ethylhexylureido)toluene (EHUT), which self-assembles into supramolecular structures (tubes and filaments) in apolar solvents dodecane and cyclohexane. High levels of pressure (roughly above 20 MPa) are found to slow down the terminal relaxation process; however, the increases in the entanglement plateau modulus and the associated persistence length are not significant. The concentration dependence of the plateau modulus, relaxation times (fast and slow), and correlation length is practically the same for all pressures and exhibits distinct power-law behavior in different regimes. Within the tube phase in dodecane, the relative viscosity increment is weakly enhanced with increasing pressure and reaches a plateau at about 60 MPa. In fact, depending on concentration, the application of pressure in the tube regime may lead to a transition from a viscous (unentangled) to a viscoelastic (partially entangled to well-entangled) solution. For well-entangled, long tubes, the extent of the plateau regime (ratio of high- to low-moduli crossover frequencies) increases with pressure. The collective information from these observations is summarized in a temperature-pressure state diagram. These findings provide ingredients for the formulation of a solid theoretical framework to better understand and exploit the role of pressure in the structure and dynamics of supramolecular polymers.