In this paper we demonstrate how broadband cavity enhanced absorption spectroscopy (CEAS) with supercontinuum (SC) radiation in the near-infrared spectral range can be used as a sensitive, multiplexed, and simple tool to probe gas-phase species in high-temperature environments. Near-infrared SC radiation is generated by pumping a standard single-mode fiber with a picosecond fiber laser. Standard low reflectivity mirrors are used for the cavity and an optical spectrum analyzer is used for the detection of gas-phase species in combustion. The method is demonstrated by measuring flame generated H2O in the 1500 to 1550 nm region and room-temperature CO2 between 1520 nm and 1660 nm. The broadband nature of the technique permits hundreds of rotational features to be recorded, giving good potential to unravel complex, convoluted spectra. We discuss practical issues concerning the implementation of the technique and present a straightforward method for calibration of the CEAS system via a cavity ringdown measurement. Despite the large spectral variation of SC radiation from pulse to pulse, it is shown that SC sources can offer good stability for CEAS where a large number of SC pulses are typically averaged.