It has been known that all bipartite pure quantum states can be certified by quantum self-testing, i.e., any such states can be pinned down completely based on the statistics produced by untrusted local quantum measurements. In this article, we prove that in the semi-device-independent setting where the underlying Hilbert space dimension is known beforehand, an arbitrary d×d bipartite pure state can be self-tested by correlations generated with a single measurement on each party, where each measurement yields only 2d outcomes. The number of outcomes could further be lowered to d+1 for a dense set of bipartite pure states. This contrasts with the fully device-independent scenario, where quantum nonlocality is necessary and each party conducts at least two different quantum measurements to produce the desired correlation. We also prove the quantitative robustness of the protocol with d + 1 outcomes to quantum noises and experimental imperfections. This result could offer significant convenience when certifying bipartite quantum states using untrusted quantum devices in future quantum industries.