We report an unanticipated transformation from a non-Fermi-to a Fermi-liquid state driven by magnetic field in a highly overdoped Tl2Ba2CuO6+x with T-c approximate to 15 K. From the c-axis resistivity measured up to 45 T, we show that the Fermi-liquid rho(c) = rho(c)(0) + AT(2) features, accompanied by a field-linear magnetoresistance, appear above a temperature-dependent field H-FL, which decreases linearly with decreasing temperature and points to a quantum critical point near the upper critical field H-c2(0). The observed field-induced quantum criticality with a power-law diverging A(H) bears a striking resemblance to that of heavy-fermion superconductor CeCoIn5, suggesting a common underlying physics in these strongly correlated electron systems.