We are interested in solving the unique continuation problem for the heat equation, i.e., we want to reconstruct the solution of the heat equation given its (noised) value in a subset of the computational domain. Both initial and boundary data can be unknown. We discretize this problem using a space-time discontinuous Galerkin method (including hybrid variables in space) and look for the solution that minimizes a discrete Lagrangian. We establish discrete inf-sup stability and bound the consistency error, leading to a priori estimates on the residual. Our main result, proving the convergence of the discrete solution to the exact solution, combines this a priori bound with a conditional stability estimate at the continuous level. The rate of convergence depends on the conditional stability, the approximation order in space and in time, and the size of the perturbations in data. Quite importantly, the weight of the regularization term depends on the time-step and the mesh-size, and we show how to choose it to preserve the best possible decay rates on the error. Finally, we run numerical simulations to assess the performance of the method in practice.