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The accumulation of noise in quantum computers is the dominant issue stymieing the push of quantum algorithms beyond their classical counterparts. We do not expect to be able to afford the overhead required for quantum error correction in the next decade, so in the meantime we must rely on low-cost, unscalable error mitigation techniques to bring quantum computing to its full potential. This paper presents a new error mitigation technique based on quantum phase estimation that can also reduce errors in expectation value estimation (e.g., for variational algorithms). The general idea is to apply phase estimation while effectively post-selecting for the system register to be in the starting state, which allows us to catch and discard errors which knock us away from there. We refer to this technique as "verified phase estimation" (VPE) and show that it can be adapted to function without the use of control qubits in order to simplify the control circuitry for near-term implementations. Using VPE, we demonstrate the estimation of expectation values on numerical simulations of intermediate scale quantum circuits with multiple orders of magnitude improvement over unmitigated estimation at near-term error rates (even after accounting for the additional complexity of phase estimation). Our numerical results suggest that VPE can mitigate against any single errors that might occur; i.e., the error in the estimated expectation values often scale as O(p^2), where p is the probability of an error occurring at any point in the circuit. This property, combined with robustness to sampling noise reveal VPE as a practical technique for mitigating errors in near-term quantum experiments.