The ability to resolve multiple fluorescent emissions from different biological targets in video rate applications, such as endoscopy and intraoperative imaging, has traditionally been limited by the use of filter-based imaging systems. Hyperspectral imaging (HSI) facilitates the detection of both spatial and spectral information in a single data acquisition, however, instrumentation for HSI is typically complex, bulky and expensive. We sought to overcome these limitations using a novel robust and low cost HSI camera based on a spectrally resolved detector array (SRDA). We integrated this HSI camera into a wide-field reflectance-based imaging system operating in the near-infrared range to assess the suitability for $in\hspace{0.25em}vivo$ imaging of exogenous fluorescent contrast agents. Using this fluorescence HSI (fHSI) system, we were able to accurately resolve the presence and concentration of at least 7 fluorescent dyes in solution. We also demonstrate high spectral unmixing precision, signal linearity with dye concentration and at depth in tissue mimicking phantoms, and delineate 4 fluorescent dyes $in\hspace{0.25em}vivo$. Our approach, including statistical background removal, could be directly generalised to broader spectral ranges, for example, to resolve tissue reflectance or autofluorescence and in future be tailored to video rate applications requiring snapshot HSI data acquisition.