This paper presents an averaging procedure for improving the Euler and Heun methods of numerical approximation of the solution of the vector random ordinary differential equation \[ \frac{dx}{dt}=G(t,\omega) +g(t,\omega) H(x,\omega), \] where \(G\), \(g\) are Hölder continuous in \(t\), and \(H\) is once (for Euler) or twice (for Heun) continuously differentiable in \(x\). It is shown that by averaging \(G\) and \(g\) over \(N\) equally spaced values of \(t\) in each discretization subinterval, order of convergence 1 for the Euler method and 2 for the Heun method can be attained, whereas the standard Euler and Heun methods may have only fractional order. Two examples are given which demonstrate that the averaging procedure approximations can show a substantial reduction in error over the approximations generated by the standard Euler and Heun methods.