We present a Monte Carlo algorithm that allows a small length scale numerical study of DNA gel electrophoresis in high electric fields, similar to the fluctuating bond model for dynamical properties of polymeric systems. This approach combines advantages of lattice Monte Carlo methods with those from continuous Brownian dynamics algorithms, and also takes into account the persistence length of DNA, as well as the random nature of the gel. The initial orientation and acceleration of a random-walk DNA conformation shows a number of features that can be related to experimental results. The detailed description of DNA motion provided by this approach may lead to a first realistic computer study of the process of DNA sequencing.