Acceptor-sensitized quantitative Förster resonance energy transfer (FRET) measurement (E-FRET) is mainly impeded by donor emission crosstalk and acceptor direct excitation crosstalk. In this paper, we develop a novel E-FRET approach (Lux-E-FRET) based on linear unmixing (Lux) of the fluorescence intensity ratio between two detection channels with each excitation of two different wavelengths. The two detection channels need not to selectively collect the emission of donor or acceptor, and the excitation wavelengths need not to selectively excite donor or acceptor. For a tandem FRET sensor, Lux-E-FRET only needs single excitation wavelength. We performed Lux-E-FRET measurements on our dual-channel wide-field fluorescence microscope for FRET constructs in living cells, and obtained consistent FRET efficiencies with those measured by other methods. Collectively, Lux-E-FRET completely overcomes all spectral crosstalks and thus is applicable to the donor-acceptor pair with larger spectral overlapping.