A nonequilibrium, bistable flow driven by exponentially correlated Gaussian noise is considered. An approximate, nonlinear Fokker-Planck-type equation, modeling effectively the long-time dynamics of the bistable, non-Markovian flow is constructed, and the mean sojourn time is evaluated in the limit of weak noise. Keeping the noise strength constant, the mean sojourn time is predicted to undergo an exponential increase with increasing noise-correlation time. Representing the bistable, colored noise dynamics with an electronic circuit, the mean of the sojourn time and the sojourn-time distribution have been measured experimentally. The experiments confirm the exponential increase for the mean sojourn time and close quantitative agreement with this newly proposed theoretical approach is found. In contrast, previous approximation schemes which expand around the Markovian theory (zero-noise correlation time) would predict for this case an exponentially decreasing mean sojourn time upon increasing the noise-correlation time, in marked disagreement with the present measurements.