In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels at the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting regularity and additional low-rank structure of the solution. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.