Considering autonomous searching for an unknown source in complex hazardous environments while balancing search efficiency and flight safety, this paper proposes a framework called source seeking with cumulative exposure minimization (SSCEM) for an unmanned aerial vehicle (UAV) equipped with gas sensors. First, the hazardous substance dispersion is formulated by using an isotropic plume model, and the stochastic sensor measurement in turbulent environments is modeled as a Poisson distribution. Second, the source term including the source location and release rate, is estimated by using a Bayesian inference framework. A particle filter is adopted to reduce the computational burden. Third, as cumulative exposure to hazardous fields can lead to airborne device failures and endanger flight safety, to achieve a reasonable trade-off among exploitation, exploration and flight safety, SSCEM for UAVs is proposed based on information theory and an interfered fluid dynamical system (IFDS), where heuristic action candidates in a continuous domain and a cost function considering cumulative exposure collectively drive the UAV to approach the unknown source location while reducing cumulative exposure, and no-fly zone avoidance is realized using the IFDS. Eventually, source seeking is efficiently performed by the UAV with no-fly zone avoidance and kinematic requirements being met, and comparative simulation results demonstrate the performance of SSCEM.