Abstract Attempts to monitor ocean eddy heat transport are strongly limited by the sparseness of available observations and the fact that heat transport is a quadratic, sign-indefinite quantity that is particularly sensitive to unresolved scales. In this article, a suite of stochastic filtering strategies for estimating eddy heat transport are tested in idealized two-layer simulations of mesoscale oceanic turbulence at high and low latitudes under a range of observation scenarios. A novel feature of these filtering strategies is the use of computationally inexpensive stochastic models to forecast the underlying nonlinear dynamics. The stochastic model parameters can be estimated by regression fitting to climatological energy spectra and correlation times or by adaptively learning these parameters “on-the-fly” from the observations themselves. The authors show that, by extracting high-wavenumber information that has been aliased into the low wavenumber band, “stochastically super-resolved” velocity fields with a nominal resolution increase of a factor of 2 or more can be derived. Observations of the upper-layer streamfunction are projected onto an empirical orthogonal function basis for the vertical structure to produce filtered estimates for both upper- and lower-layer streamfunctions and hence net heat transport. The resulting time-mean poleward eddy heat transport is significantly closer to the true value when compared with standard estimates based upon optimal interpolation. By contrast, the temporal variability of the heat transport is underestimated because of poor temporal resolution. Implications for estimating poleward eddy heat transport using current and next-generation altimeters are discussed.