We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference efficiently on models that have very large treewidth. The highlight of the advance is that it works with standard (dense) factors, without the need for sparse factors or techniques based on knowledge compilation that are commonly utilized. This is significant as it permits a direct implementation of the improved variable elimination algorithm using tensors and their operations, leading to extremely efficient implementations especially when learning model parameters. We illustrate the efficacy of our proposed algorithm by compiling Bayesian network queries into tensor graphs and then learning their parameters from labeled data using a standard tool for tensor computation.