We report the discovery of extended X-ray emission within the young star cluster NGC602a in the Wing of the Small Magellanic Cloud (SMC) based on observations obtained with the Chandra X-ray Observatory. X-ray emission is detected from the cluster core area with the highest stellar density and from a dusty ridge surrounding the HII region. We use a census of massive stars in the cluster to demonstrate that a cluster wind or wind-blown bubble is unlikely to provide a significant contribution to the X-ray emission detected from the central area of the cluster. We therefore suggest that X-ray emission at the cluster core originates from an ensemble of low- and solar-mass pre-main-sequence (PMS) stars, each of which would be too weak in X-rays to be detected individually. We attribute the X-ray emission from the dusty ridge to the embedded tight cluster of the new-born stars known in this area from infrared studies. Assuming that the levels of X-ray activity in young stars in the low-metallicity environment of NGC 602a are comparable to their Galactic counterparts, then the detected spatial distribution, spectral properties, and level of X-ray emission are largely consistent with those expected from low- and solar-mass PMS stars and young stellar objects (YSOs). This is the first discovery of X-ray emission attributable to PMS stars and YSOs in the SMC, which suggests that the accretion and dynamo processes in young, low-mass objects in the SMC resemble those in the Galaxy.