Unraveling latent structure by means of multilinear models of tensor data is of paramount importance in timely inference tasks encountered with 'Big Data' analytics. However, increasingly noisy, heterogeneous, and incomplete datasets as well as the need for real-time processing of streaming data pose major challenges to this end. The present paper introduces a novel online (adaptive) algorithm to decompose low-rank tensors with missing entries, and perform imputation as a byproduct. The novel estimator minimizes an exponentially- weighted least-squares fitting error along with a separable regularizer of the PARAFAC decomposition factors, to trade-off fidelity for complexity of the approximation captured by the decomposition's rank. Leveraging stochastic gradient descent iterations, a scalable, real-time algorithm is developed and its convergence is established under simplifying technical assumptions. Simulated tests with cardiac magnetic resonance imagery (MRI) data confirm the efficacy of the proposed algorithm in imputing up to 75% missing entries.