The application of monocular dense Simultaneous Localization and Mapping (SLAM) is often hindered by high latency, large GPU memory consumption, and reliance on camera calibration. To relax this constraint, we propose EC3R-SLAM, a novel calibration-free monocular dense SLAM framework that jointly achieves high localization and mapping accuracy, low latency, and low GPU memory consumption. This enables the framework to achieve efficiency through the coupling of a tracking module, which maintains a sparse map of feature points, and a mapping module based on a feed-forward 3D reconstruction model that simultaneously estimates camera intrinsics. In addition, both local and global loop closures are incorporated to ensure mid-term and long-term data association, enforcing multi-view consistency and thereby enhancing the overall accuracy and robustness of the system. Experiments across multiple benchmarks show that EC3R-SLAM achieves competitive performance compared to state-of-the-art methods, while being faster and more memory-efficient. Moreover, it runs effectively even on resource-constrained platforms such as laptops and Jetson Orin NX, highlighting its potential for real-world robotics applications.