Ethereum smart contracts are automated decentralized applications on the blockchain that describe the terms of the agreement between buyers and sellers, reducing the need for trusted intermediaries and arbitration. However, the deployment of smart contracts introduces new attack vectors into the cryptocurrency systems. In particular, programming flaws in smart contracts can be and have already been exploited to gain enormous financial profits. It is thus an emerging yet crucial issue to detect vulnerabilities of different classes in contracts in an efficient manner. Existing machine learning-based vulnerability detection methods are limited and only inspect whether the smart contract is vulnerable, or train individual classifiers for each specific vulnerability, or demonstrate multi-class vulnerability detection without extensibility consideration. To overcome the scalability and generalization limitations of existing works, we propose ESCORT, the first Deep Neural Network (DNN)-based vulnerability detection framework for Ethereum smart contracts that support lightweight transfer learning on unseen security vulnerabilities, thus is extensible and generalizable. ESCORT leverages a multi-output NN architecture that consists of two parts: (i) A common feature extractor that learns the semantics of the input contract; (ii) Multiple branch structures where each branch learns a specific vulnerability type based on features obtained from the feature extractor. Experimental results show that ESCORT achieves an average F1-score of 95% on six vulnerability types and the detection time is 0.02 seconds per contract. When extended to new vulnerability types, ESCORT yields an average F1-score of 93%. To the best of our knowledge, ESCORT is the first framework that enables transfer learning on new vulnerability types with minimal modification of the DNN model architecture and re-training overhead.