Monolayered, semiconducting molybdenum disulfide (MoS2) is of considerable interest for its potential applications in next-generation flexible, wearable, and transparent photodetectors because it has outstanding physical properties coupled with unique atomically thin dimensions. However, there is still a lack of understanding in terms of the underlying mechanisms responsible for the photoresponse dynamics, which makes it difficult to identify the appropriate device design strategy for achieving a fast photoresponse time in MoS2 photodetectors. In this study, we investigate the importance of surface functionalization on controlling the charge carrier densities in a MoS2 monolayer and in turn the corresponding behavior of the photoresponse in relation to the position of the Fermi-level and the energy band structure. We find that the p-doping and n-doping, which is achieved through the surface functionalization of the MoS2 monolayer, leads to devices with different photoresponse behavior. Specifically, the MoS2 devices with surface functional groups contributing to p-doping exhibited a faster response time as well as higher sensitivity compared to that observed for the MoS2 devices with surface functional groups contributing to n-doping. We attribute this difference to the degree of bending in the energy bands at the metal-semiconductor junction as a result of shifting in the Fermi-level position, which influences the optoelectronic transport properties as well as the recombination dynamics leading to a low dark and thus high detectivity and fast decay time. Based upon these findings, we have also demonstrated the broad applicability of surface functionalization by fabricating a flexible MoS2 photodetector that shows an outstanding decay time of 0.7 s, which is the fastest response time observed in flexible MoS2 detectors ever reported.