With the emergence of large language models (LLMs), investigating if they can surpass humans in areas such as emotion recognition and empathetic responding has become a focal point of research. This paper presents a comprehensive study exploring the empathetic responding capabilities of four state-of-the-art LLMs: GPT-4, LLaMA-2-70B-Chat, Gemini-1.0-Pro, and Mixtral-8x7B-Instruct in comparison to a human baseline. We engaged 1,000 participants in a between-subjects user study, assessing the empathetic quality of responses generated by humans and the four LLMs to 2,000 emotional dialogue prompts meticulously selected to cover a broad spectrum of 32 distinct positive and negative emotions. Our findings reveal a statistically significant superiority of the empathetic responding capability of LLMs over humans. GPT-4 emerged as the most empathetic, marking approximately 31% increase in responses rated as "Good" compared to the human benchmark. It was followed by LLaMA-2, Mixtral-8x7B, and Gemini-Pro, which showed increases of approximately 24%, 21%, and 10% in "Good" ratings, respectively. We further analyzed the response ratings at a finer granularity and discovered that some LLMs are significantly better at responding to specific emotions compared to others. The suggested evaluation framework offers a scalable and adaptable approach for assessing the empathy of new LLMs, avoiding the need to replicate this study's findings in future research.