In the search for materials for quantum information science applications, colloidal semiconductor nanoplatelets (NPLs) have emerged as a highly promising new class of materials due to their interesting optical properties, such as narrow emission linewidth and fast photoluminescence (PL) lifetimes at room temperature. So far only few works focused on the quantum properties of their emission, however, NPLs, with their atomic-scale thickness and one-dimensional quantum confinement, are promising candidates for single-photon sources. Here, we demonstrate room-temperature single-photon emission from core/shell CdSe/ZnS NPLs, which feature 8$x$20 nm$^2$ surface area and 1 nm shell. The limited surface area ensures effective Auger non-radiative recombination, resulting in highly efficient single-photon generation with values of photon purity as low as $g^{(2)}(\tau)=0.04$. The observed long-period blinking and bleaching, typical of such thin shells, can be easily reduced by increasing the shell thickness. This work establishes NPLs as new single-photon sources very well suited for integration into quantum photonic systems.