Two-dimensional transition metal dichalcogenides (TMDCs) like MoS2 are promising candidates for various optoelectronic applications. The typical photoluminescence (PL) of monolayer MoS2 is however known to suffer very low quantum yields. We demonstrate a 10-fold increase of MoS2 excitonic PL enabled by nonradiative energy transfer (NRET) from adjacent nanocrystal quantum dot (NQD) films. The understanding of this effect is facilitated by our application of transient absorption (TA) spectroscopy to monitor the energy influx into the monolayer MoS2 in the process of ET from photoexcited CdSe/ZnS nanocrystals. In contrast to PL spectroscopy, TA can detect even non-emissive excitons, and we register an order of magnitude enhancement of the MoS2 excitonic TA signatures in hybrids with NQDs. The appearance of ET-induced nanosecond-scale kinetics in TA features is consistent with PL dynamics of energy-accepting MoS2 and PL quenching data of the energy-donating NQDs. The observed enhancement is attributed to the reduction of recombination losses for excitons gradually transferred into MoS2 under quasi-resonant conditions as compared with their direct photoproduction. The TA and PL data clearly illustrate the efficacy of MoS2 and likely other TMDC materials as energy acceptors and the possibility of their practical utilization in NRET-coupled hybrid nanostructures.