Coordinating heterogeneous multi-robot systems (MRS) for complex, long-horizon tasks requires both flexible high-level reasoning and efficient low-level scheduling. Existing LLM-based approaches address the reasoning side but introduce two critical bottlenecks: (1) repeated LLM inference during execution, which inflates latency with agent count, and (2) offline, pre-committed scheduling, which forces robots to idle while waiting for sequentially ordered predecessors even when independent work is available. This paper presents OSDAG, a novel framework that integrates LLM-based task reasoning with Directed Acyclic Graph (DAG) representation and constraint-aware online scheduling. The LLM is invoked once to decompose a natural-language instruction into a dependency-annotated task graph, and a lightweight online scheduler then allocates ready tasks to idle agents in real time. The DAG representation encodes both precedence and resource constraints, ensuring correctness while exposing all available parallelism. Experiments across five benchmark scenarios demonstrate that OSDAG achieves 5-15x faster reasoning time compared to dialogue-based methods, reduces makespan by up to 38% over sequential baselines, and maintains competitive success rates. Both simulation and real-world experiments on dual-arm manipulation tasks validate the effectiveness and practicality of the proposed approach for efficient multi-robot coordination. The website and resources are available at http://thanhnguyencanh.github.io/LLM_DAG4MultiRobot