Previous single-particle spectroscopic studies of colloidal quantum dots have indicated a significant spread in biexciton lifetimes across an ensemble of nominally identical nanocrystals. It has been speculated that in addition to dot-to-dot variation in physical dimensions, this spread is contributed to by variations in the structure of the quantum dot interface, which controls the shape of the confinement potential. Here we directly evaluate the effect of the composition of the core-shell interface on single- and multi-exciton dynamics via side-by-side measurements of individual core-shell CdSe-CdS nanocrystals with a sharp vs. smooth (graded) interface. To realize the latter type of structures, we incorporate a CdSexS1-x alloy layer of controlled composition and thickness between the CdSe core and the CdS shell. We observe that while having essentially no effect on single-exciton decay, the interfacial alloy layer leads to a systematic increase in biexciton lifetimes. This observation provides direct experimental evidence that in addition to the size of the quantum dot, its interfacial properties also significantly affect the rate of Auger recombination, which governs biexciton decay. These findings help rationalize previous observations of a significant heterogeneity in the biexciton lifetimes across similarly sized quantum dots and should facilitate the development of Auger-recombination-free colloidal nanostructures for a range of applications from lasers and light-emitting diodes to photodetectors and solar cells.