In this work, we propose a collision-free source-seeking control framework for a unicycle robot traversing an unknown cluttered environment. In this framework, obstacle avoidance is guided by the control barrier functions (CBF) embedded in quadratic programming, and the source-seeking control relies solely on the use of onboard sensors that measure the signal strength of the source. To tackle the mixed relative degree and avoid the undesired position offset for the nonholonomic unicycle model, we propose a novel construction of a control barrier function (CBF) that can directly be integrated with our recent gradient-ascent source-seeking control law. We present a rigorous analysis of the approach. The efficacy of the proposed approach is evaluated via Monte-Carlo simulations, as well as, using a realistic dynamic environment with moving obstacles in Gazebo/ROS.