This work is concerned with the computation of the action of a matrix function f(A), such as the matrix exponential or the matrix square root, on a vector b. For a general matrix A, this can be done by computing the compression of A onto a suitable Krylov subspace. Such compression is usually computed by forming an orthonormal basis of the Krylov subspace using the Arnoldi method. In this work, we propose to compute (nonorthonormal) bases in a faster way and to use a fast randomized algorithm for least -squares problems to compute the compression of A onto the Krylov subspace. We present some numerical examples which show that our algorithms can be faster than the standard Arnoldi method while achieving comparable accuracy.