Accurate estimation of origin-to-destination (OD) traffic flows provides valuable input for network management tasks. However, lack of flow-level observations as well as intentional and unintentional anomalies pose major challenges toward achieving this goal. Leveraging the low intrinsic- dimensionality of OD flows and the sparse nature of anomalies, this paper proposes a convex program with nuclear-norm and ℓ1-norm regularization terms to estimate the nominal and anomalous traffic components, using a small subset of (possibly anomalous) flow counts in addition to link counts. Analysis and simulations confirm that the said estimator can exactly recover sufficiently low-dimensional nominal traffic and sparse enough anomalies when the routing matrix is column-incoherent, and an adequate amount of flow counts are randomly sampled. The results offer valuable insights about the measurement types and network scenaria giving rise to accurate traffic estimation. Tests with real Internet data corroborate the effectiveness of the novel estimator.