The azimuthal anisotropy of mesons in high-multiplicity proton-lead collisions is studied using data collected by the CMS experiment at a nucleon-nucleon center-of-mass energy of 8.16 TeV. The mesons are reconstructed using their dimuon decay channel. The anisotropy is characterized by the second Fourier harmonic coefficients, found using a two-particle correlation technique, in which the mesons are correlated with charged hadrons. A large pseudorapidity gap is used to suppress short-range correlations. Nonflow contamination from the dijet background is removed using a low-multiplicity subtraction method, and the results are presented as a function of transverse momentum. The azimuthal anisotropies are smaller than those found for charmonia in proton-lead collisions at the same collision energy, but are consistent with values found for mesons in lead-lead interactions at a nucleon-nucleon center-of-mass energy of 5.02 TeV.