The paper proposes a general framework to study the exponential stabilization problem for a class of nonlinear systems. By using the σ-processing, the exponential stabilization problem can be solved provided that all solutions of an augmented system are uniformly globally bounded. Based on this result, a simple and general stability criterion is presented. two well-known nonholonomic systems are shown that they fall into the considered class and satisfy the proposed criterion. In particular, the exponential stability can be achieved for these systems. These examples validate the effectiveness of the proposed results.