Tensor-based fusion that couples the high spatial resolution of a multispectral image (MSI) to the high spectral resolution of a hyperspectral image (HSI) is considered. The fusion problem is first formulated mathematically as a convex optimization of a tensor trace norm imposing low-rank spatially as well as spectrally, with an alternating-directions optimization featuring linearization providing the solution. Although prior tensor-based fusion approaches typically resort to tensor decomposition, the proposed algorithm exploits ideas from the field of tensor completion to directly impose a low-rank property spatially and spectrally while avoiding the computationally complex patch clustering and dictionary learning common to competing fusion techniques. Additionally, small modifications to the basic optimization permit a fusion process robust to missing hyperspectral values such as those that can result from dead stripes in real hyperspectral sensors. The experimental evaluations on both synthetic imagery as well as real imagery demonstrate that the resulting low-rank tensor-approximation (LRTA) fusion algorithm preserves both spatial details and texture, yielding significantly improved image quality when compared to other state-of-the-art fusion methods as well as effective restoration under conditions of missing stripes within the HSI.