The study of third-order statistics in large-scale structure analyses has been hampered by the increased complexity of bispectrum estimators (compared to power spectra), the large dimensionality of the data vector, and the difficulty in estimating its covariance matrix. In this paper we present the filtered-squared bispectrum (FSB), an estimator of the projected bispectrum effectively consisting of the cross-correlation between the square of a field filtered on a range of scales and the original field. Within this formalism, we are able to recycle much of the infrastructure built around power spectrum measurement to construct an estimator that is both fast and robust against mode-coupling effects caused by incomplete sky observations. Furthermore, we demonstrate that the existing techniques for the estimation of analytical power spectrum covariances can be used within this formalism to calculate the bispectrum covariance at very high accuracy, naturally accounting for the most relevant Gaussian and non-Gaussian contributions in a model-independent manner.