Traditional knowledge-based situation awareness (SA) modes struggle to adapt to the escalating complexity of today's Energy Internet of Things (EIoT), necessitating a pivotal paradigm shift. In response, this work introduces a pioneering data-driven SA framework, termed digital twin-based situation awareness (DT-SA), aiming to bridge existing gaps between data and demands, and further to enhance SA capabilities within the complex EIoT landscape. First, we redefine the concept of digital twin (DT) within the EIoT context, aligning it with data-intensive scientific discovery paradigm (the Fourth Paradigm) so as to waken EIoT's sleeping data; this contextual redefinition lays the cornerstone of our DT-SA framework for EIoT. Then, the framework is comprehensively explored through its four fundamental steps: digitalization, simulation, informatization, and intellectualization. These steps initiate a virtual ecosystem conducive to a continuously self-adaptive, self-learning, and self-evolving big model (BM), further contributing to the evolution and effectiveness of DT-SA in engineering. Our framework is characterized by the incorporation of system theory and Fourth Paradigm as guiding ideologies, DT as data engine, and BM as intelligence engine. This unique combination forms the backbone of our approach. This work extends beyond engineering, stepping into the domain of data science -- DT-SA not only enhances management practices for EIoT users/operators, but also propels advancements in pattern analysis and machine intelligence (PAMI) within the intricate fabric of a complex system. Numerous real-world cases validate our DT-SA framework.