Text-video retrieval, a prominent sub-field within the domain of multimodal information retrieval, has witnessed remarkable growth in recent years. However, existing methods assume video scenes are consistent with unbiased descriptions. These limitations fail to align with real-world scenarios since descriptions can be influenced by annotator biases, diverse writing styles, and varying textual perspectives. To overcome the aforementioned problems, we introduce WAVER, a cross-domain knowledge distillation framework via vision-language models through open-vocabulary knowledge designed to tackle the challenge of handling different writing styles in video descriptions. WAVER capitalizes on the open-vocabulary properties that lie in pre-trained vision-language models and employs an implicit knowledge distillation approach to transfer text-based knowledge from a teacher model to a vision-based student. Empirical studies conducted across four standard benchmark datasets, encompassing various settings, provide compelling evidence that WAVER can achieve state-of-the-art performance in text-video retrieval task while handling writing-style variations.