Compared to conventional decomposition methods that use ellipses or polygons to represent free space, starshaped representation can better capture the natural distribution of sensor data, thereby exploiting a larger portion of traversable space. This paper introduces a novel motion planning and control framework for navigating robots in unknown and cluttered environments using a dynamically constructed starshaped roadmap. Our approach generates a starshaped representation of the surrounding free space from real-time sensor data using piece-wise polynomials. Additionally, an incremental roadmap maintaining the connectivity information is constructed, and a searching algorithm efficiently selects short-term goals on this roadmap. Importantly, this framework addresses dead-end situations with a graph updating mechanism. To ensure safe and efficient movement within the starshaped roadmap, we propose a reactive controller based on Dynamic System Modulation (DSM). This controller facilitates smooth motion within starshaped regions and their intersections, avoiding conservative and short-sighted behaviors and allowing the system to handle intricate obstacle configurations in unknown and cluttered environments. Comprehensive evaluations in both simulations and real-world experiments show that the proposed method achieves higher success rates and reduced travel times compared to other methods. It effectively manages intricate obstacle configurations, avoiding conservative and myopic behaviors.