As conversational models become increasingly available to the general public, users are engaging with this technology in social interactions. Such unprecedented interaction experiences may pose considerable social and psychological risks to the users unless the technology is properly controlled. This highlights the need for scalable and robust evaluation metrics for conversational chatbots. Existing evaluation metrics aim to automate offline user evaluation and approximate human judgment of pre-curated dialogs. However, they are limited in their ability to capture subjective perceptions of users who actually interact with the bots and might not generalize to real-world settings. To address this limitation, we propose an approach to approximate online human evaluation leveraging large language models (LLMs) from the GPT family. We introduce a new Dialog system Evaluation framework based on Prompting (DEP), which enables a fully automatic evaluation pipeline that replicates live user studies and achieves an impressive correlation with human judgment (up to Pearson r=0.95 on a system level). The DEP approach involves collecting synthetic chat logs of evaluated bots with an LLM in the other-play setting, where the LLM is carefully conditioned to follow a specific scenario. We further explore different prompting approaches to produce evaluation scores with the same LLM. The best performing prompts, which contain few-shot demonstrations and instructions, show outstanding performance on the tested dataset and demonstrate the ability to generalize to other dialog corpora.