We present an approach to estimate the operational distinguishability between an entangled state and any separable state directly from measuring an entanglement witness. We show that this estimation also implies bounds on a variety of other well-known entanglement quantifiers. This approach for entanglement estimation is then extended to both the measurement-device-independent scenario and the fully device-independent scenario, where we obtain nontrivial but suboptimal bounds. The procedure requires no numerical optimization and is easy to compute. It offers ways for experimenters to not only detect, but also quantify, entanglement from the standard entanglement witness procedure.