The space-air-ground integrated network (SAGIN) is regarded as a promising approach for providing ubiquitous Internet access anytime and anywhere. With virtualization technologies and multi-access edge computing, data transmission and data processing in SAGINs are abstracted as services. Space-air-ground service computing flexibly integrates and manages these services in SAGINs based on service-oriented architecture. However, it is significant but very challenging to provide Internet of Things service with high QoS in space-air-ground service computing due to the distributed service management, and the mobility of both infrastructures and users. Therefore, in this article, we investigate service coordination to guarantee the QoS in space-air-ground service computing. In particular, we first introduce three service coordination scenarios: fine-grained, medium-grained, and coarse-grained service coordination. Then we design a service coordination framework that contains three tiers: edge node tier, service function routing tier, and global control tier. After that, we propose a service coordination approach to reduce the service delay at low cost, which considers the selection with foresight and updates based on threshold. Experimental results show the advantages of our service coordination approach in terms of service delay and cost.