To bridge the gap between limited hardware access and the huge demand for experiments for Noisy Intermediate-Scale Quantum (NISQ) computing system study, a simulator which can capture the modeling of both the quantum processor and its classical control system to realize early-stage evaluation and design space exploration, is naturally invoked but still missing. This paper presents SANQ, a Simulation framework for Architecting NISQ computing system. SANQ consists of two components, 1) an optimized noisy quantum computing (QC) simulator with flexible error modeling accelerated by eliminating redundant computation, and 2) an architectural simulation infrastructure to construct behavior models for evaluating the control systems. SANQ is validated with existing NISQ quantum processor and control systems to ensure simulation accuracy. It can capture the variance on the QC device and simulate the timing behavior precisely (<1% and 10% error for various real control systems). Several potential applications are proposed to show that SANQ could benefit the future design of NISQ compiler, architecture, etc.