Strongly disordered superconductors in a magnetic field display many characteristic properties of type-II superconductivity— except at low temperatures where an anomalous linear T-dependence of the resistive critical field Bc2 is routinely observed. This behavior violates the conventional theory of superconductivity, and its origin remains a long-standing puzzle. In a combined experimental and theoretical effort, we conducted systematic measurements of the critical magnetic field and current on amorphous indium oxide films of various levels of disorder. Surprisingly, our measurements show that the Bc2 anomaly near zero-temperature is accompanied by a clear mean-field like scaling behavior of the critical current. Theoretically, we show that these are consequences of the vortex-glass ground state and its thermal fluctuations. This theory further predicts the linear-T anomaly to occur in films as well as bulk superconductors with a slope that depends on the normal-state sheet resistance—in agreement with experimental data. Thus, our combined study reveals universal low-temperature behavior of Bc2 in a large class of disordered superconductors.