We use a quadratic maximum likelihood (QML) method to estimate the angular power spectrum of the cross-correlation between cosmic microwave background and large-scale structure maps as well as their individual auto-spectra. We describe our implementation of this method and demonstrate its accuracy on simulated maps. We apply this optimal estimator to Wilkinson Microwave Anisotropy Probe (WMAP) 7-yr and National Radio Astronomical Observatory (NRAO) Very Large Array Sky Survey (NVSS) data and explore the robustness of the angular power spectrum estimates obtained by the QML method. With the correction of the declination systematics in NVSS, we can safely use most of the information contained in this survey. We then make use of the angular power spectrum estimates obtained by the QML method to derive constraints on the dark energy critical density in a flat Λ cold dark matter model by different likelihood prescriptions. When using just the cross-correlation between WMAP 7-yr and NVSS maps with 1°.8 resolution, the best-fitting model has a cosmological constant of approximately 70 per cent of the total energy density, disfavouring an Einstein–de sitter universe at more than 2σ confidence level.