We present a novel method for the optimal selection of quasars using time-series observations in a single photometric bandpass. Utilizing the damped random walk model of Kelly et al., we parameterize the ensemble quasar structure function in Sloan Stripe 82 as a function of observed brightness. The ensemble model fit can then be evaluated rigorously for and calibrated with individual light curves with no parameter fitting. This yields a classification in two statistics—one describing the fit confidence and the other describing the probability of a false alarm—which can be tuned, a priori, to achieve high quasar detection fractions (99% completeness with default cuts), given an acceptable rate of false alarms. We establish the typical rate of false alarms due to known variable stars as lsim3% (high purity). Applying the classification, we increase the sample of potential quasars relative to those known in Stripe 82 by as much as 29%, and by nearly a factor of two in the redshift range 2.5 < z < 3, where selection by color is extremely inefficient. This represents 1875 new quasars in a 290 deg$^2$ field. The observed rates of both quasars and stars agree well with the model predictions, with >99% of quasars exhibiting the expected variability profile. We discuss the utility of the method at high redshift and in the regime of noisy and sparse data. Our time-series selection complements well-independent selection based on quasar colors and has strong potential for identifying high-redshift quasars for Baryon Acoustic Oscillations and other cosmology studies in the LSST era.