Tensor-train (TT) decomposition has been an efficient tool to find low order approximation of large-scale, high-order tensors. Existing TT decomposition algorithms are either of high computational complexity or operating in batch-mode, hence quite inefficient for (near) real-time processing. In this paper, we propose a novel adaptive algorithm for TT decomposition of streaming tensors whose slices are serially acquired over time. By leveraging the alternating minimization framework, our estimator minimizes an exponentially weighted least-squares cost function in an efficient way. The proposed method can yield an estimation accuracy very close to the error bound. Numerical experiments show that the proposed algorithm is capable of adaptive TT decomposition with a competitive performance evaluation on both synthetic and real data.