Accurate quantification of Förster resonance energy transfer (FRET) using intensity-based methods is difficult due to the overlap of fluorophore excitation and emission spectra. Consequently, mechanisms are required to remove bleedthrough of the donor emission into the acceptor channel and direct excitation of the acceptor when aiming to excite only the donor fluorophores. Methods to circumvent donor bleedthrough using the unmixing of emission spectra have been reported, but these require additional corrections to account for direct excitation of the acceptor. Here we present an alternative method for robust quantification of FRET efficiencies based upon the simultaneous spectral unmixing of both excitation and emission spectra. This has the benefit over existing methodologies in circumventing the issue of donor bleedthrough and acceptor cross excitation without the need for additional corrections. Furthermore, we show that it is applicable with as few as two excitation wavelengths and so can be used for quantifying FRET efficiency in microscope images as easily as for data collected on a spectrofluorometer. We demonstrate the accuracy of the approach by reproducing efficiency values in well characterized FRET standards: HEK cells expressing a variety of linked cerulean and venus fluorescent proteins. Finally we describe simple ImageJ plugins that can be used to calculate and create images of FRET efficiencies from microscope images.