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Hypernasality is an abnormal resonance in human speech production, especially in patients with craniofacial anomalies such as cleft palate. In clinical application, hypernasality estimation is crucial in cleft palate diagnosis, as its results determine the subsequent surgery and additional speech therapy. Therefore, designing an automatic hypernasality assessment method will facilitate speech-language pathologists to make precise diagnoses. Existing methods for hypernasality estimation only conduct acoustic analysis based on low-resource cleft palate dataset, by using statistical or neural network-based features. In this paper, we propose a novel approach that uses automatic speech recognition model to improve hypernasality estimation. Specifically, we first pre-train an encoder-decoder framework in an automatic speech recognition (ASR) objective by using speech-to-text dataset, and then fine-tune ASR encoder on the cleft palate dataset for hypernasality estimation. Benefiting from such design, our model for hypernasality estimation can enjoy the advantages of ASR model: 1) compared with low-resource cleft palate dataset, the ASR task usually includes large-scale speech data in the general domain, which enables better model generalization; 2) the text annotations in ASR dataset guide model to extract better acoustic features. Experimental results on two cleft palate datasets demonstrate that our method achieves superior performance compared with previous approaches.