Collecting dialogue data with domain-slot-value labels for dialogue state tracking (DST) could be a costly process. In this paper, we propose a novel framework based on domain-slot related description to tackle the challenge of few-shot cross-domain DST. Specifically, we design an extraction module to extract domain-slot related verbs and nouns in the dialogue. Then, we integrates them into the description, which aims to prompt the model to identify the slot information. Furthermore, we introduce a random sampling strategy to improve the domain generalization ability of the model. We utilize a pre-trained model to encode contexts and description and generates answers with an auto-regressive manner. Experimental results show that our approaches substantially outperform the existing few-shot DST methods on MultiWOZ and gain strong improvements on the slot accuracy comparing to existing slot description methods.