We show that widely used approaches in statistical physics incorrectly indicate the existence of power-law cross-correlations between financial stock market fluctuations measured over several years and the neuronal activity of the human brain lasting for only a few minutes. While such cross-correlations are nonsensical, no current methodology allows them to be reliably discarded, leaving researchers at greater risk when the spurious nature of cross-correlations is not clear from the unrelated origin of the time series and rather requires careful statistical estimation. Here we propose a theory and method (PLCC-test) which allows us to rigorously and robustly test for power-law cross-correlations, correctly detecting genuine and discarding spurious cross-correlations, thus establishing meaningful relationships between processes in complex physical systems. Our method reveals for the first time the presence of power-law cross-correlations between amplitudes of the alpha and beta frequency ranges of the human electroencephalogram.