We use the notion of discrete vector fields to propose a new algorithm for computing the minimal free resolution of a monomial ideal. The al- gorithm consists on two steps in each of which discrete vector fields are used in a different manner. Each of the steps is based on well known constructions that can be made effective with the use of discrete vector fields. The first step consists on the construction of a resolution together with a structural discrete vector field which, at the same time, proves it is an actual resolution and provides a recursive mechanism to explicitly extend this resolution adding new elements to the ideal. The second step minimizes any monomial resolution using discrete vector fields as a way to encode usual reduction steps in a convenient way.