GeSn coatings on commercial Si substrates have gained increased interest for application in infrared detectors and lasers. The characterization of these materials is crucial for their quality assessment and in describing device performance for commercialization. The time-resolved photoluminescence is an efficient method for contactless, time-resolved characterization of GeSn optoelectronic properties. For this purpose, in this work, we developed an infrared streak camera attachment based on the broadband upconversion of infrared photoluminescence by using powerful nanosecond 1064 nm pulses. The attachment achieved picosecond time resolution, being limited by the laser pulse duration, jitter, and temporal resolution of the streak camera. The spectral range for time-resolved photoluminescence in the 1100–2400 nm interval was obtained, outperforming the range of commercial infrared InGaAs streak cameras. The developed setup was applied for excitation-dependent time-resolved photoluminescence decay measurements in a GeSn and compared to the conventional upconversion technique with the optically delayed picosecond gate pulses. The new setup provided 2D spectro-temporal images for analysis. The photoluminescence decay times in the 30–80 ps range were obtained in the GeSn layer depending on the excitation pulse energy and spectral emission wavelength. Carrier thermalization was observed as a redshift of the photoluminescence spectra with time.