6G services are evolving toward goal-oriented and AI-native communication, which are expected to deliver transformative societal benefits across various industries and promote energy sustainability. Yet today's networking architectures, built on complete decoupling of the applications and the network, cannot expose or exploit high-level goals, limiting their ability to adapt intelligently to service needs. This work introduces Goal-Oriented Multi-Agent Semantic Networking (GoAgentNet), a new architecture that elevates communication from data exchange to goal fulfilment. GoAgentNet enables applications and the network to collaborate by abstracting their functions into multiple collaborative agents, and jointly orchestrates multi-agent sensing, networking, computation, and control through semantic computation and cross-layer semantic networking, allowing the entire architecture to pursue unified application goals. We first outline the limitations of legacy network designs in supporting 6G services, based on which we highlight key enablers of our GoAgentNet design. Then, through three representative 6G usage scenarios, we demonstrate how GoAgentNet can unlock more efficient and intelligent services. We further identify unique challenges faced by GoAgentNet deployment and corresponding potential solutions. A case study on robotic fault detection and recovery shows that our GoAgentNet architecture improves energy efficiency by up to 99% and increases the task success rate by up to 72%, compared with the existing networking architectures without GoAgentNet, which underscores its potential to support scalable and sustainable 6G systems.