Generalization of synthesizer sound matching to external instrument sounds is highly challenging due to the non-differentiability of sound synthesis process which prohibits the use of out-of-domain sounds for training with synthesis parameter loss. We propose SynthRL, a novel reinforcement learning (RL)-based approach for cross-domain synthesizer sound matching. By incorporating sound similarity into the reward function, SynthRL effectively optimizes synthesis parameters without ground-truth labels, allowing fine-tuning on out-of-domain sounds. Furthermore, we introduce a transformer-based model architecture and reward-based prioritized experience replay to enhance RL training efficiency, considering the unique characteristics of the task. Experimental results demonstrate that SynthRL outperforms state-of-the-art methods on both in-domain and out-of-domain tasks. Further experimental analysis validates the effectiveness of our reward design, showing a strong correlation with human perception of sound similarity.