Fluorescence correlation spectroscopy (FCS) is a powerful tool in the time-resolved analysis of nonreacting or reacting molecules in solution, based on fluorescence intensity fluctuations. However, conventional (second-order) FCS alone is insufficient to measure all parameters needed to describe a reaction or mixture, including concentrations, fluorescence brightnesses, and forward and reverse rate constants. For this purpose, correlations of higher powers of fluorescence intensity fluctuations can be calculated to yield additional information from the single-photon data stream collected in an FCS experiment. To describe systems of diffusing and reacting molecules, considering cumulants of fluorescence intensity results in simple expressions in which the reaction and diffusion parts factorize. The computation of higher-order correlations in experiments is hindered by shot-noise and common detector artifacts, the effects of which become worse with increasing order. In this article, we introduce a technique to calculate artifact-free higher-order correlation functions with improved time resolution, and without any need for modeling and calibration of detector artifacts. The technique is formulated for general multidetector experiments and verified in both two-detector and single-detector configurations. Good signal-to-noise ratio is achieved down to 1 mu s in correlation curves up to order (2, 2). This capability makes possible a variety of new measurements including multicomponent analysis and fast reaction kinetics, as demonstrated in a companion article (10.1021/acs.jpcb.7b00408).