Modern networks have become immensely complicated, while future networks are expected to be more highly dynamic and sophisticated. In such a complex network environment, it is challenging to design effective control schemes to allocate network resources and manage network systems. Recently, Artificial Intelligence (AI) has made tremendous successes and breakthroughs. Particularly, as a branch of AI technology, the model-free experience-based machine learning (ML) technology has attracted widespread attention in the field of Network Control Problems (NCPs). Motivated by recent breakthroughs in AI technology, we share our vision of deploying ML technology to the field of solving NCPs to achieve experience-driven networking. Compared with the domain knowledge-based heuristic algorithms and fixed policies, which face mounting challenges in achieving desirable performance in highly dynamic and complicated networks, the ML-based methods can accumulate experiences by constantly collecting system feedback, and finally learn desirable/optimal control policies. In this article, we first summarize the differences between the traditional solutions and the ML-based methods and highlight the advantages of experience-driven networking with ML. Then we introduce several state-of-the-art AI-enabled experience-driven works for NCPs. In the end, we shed light on the future directions of adapting ML to more networking problems and the emerging opportunities for experience-driven networking. We hope that our work can help and encourage researchers to propose more innovative experience-driven solutions for the NCPs of modern network systems.