In E-commerce, vouchers are important marketing tools to enhance users' engagement and boost sales and revenue. The likelihood that a user redeems a voucher is a key factor in voucher distribution decision. User-item Click-Through-Rate (CTR) models are often applied to predict the user-voucher redemption rate. However, the voucher scenario involves more complicated relations among users, items and vouchers. The users' historical behavior in a voucher collection activity reflects users' voucher usage patterns, which is nevertheless overlooked by the CTR-based solutions. In this paper, we propose a Deep Multi-behavior Graph Networks (DMBGN) to shed light on this field for the voucher redemption rate prediction. The complex structural user-voucher-item relationships are captured by a User-Behavior Voucher Graph (UVG). User behavior happening both before and after voucher collection is taken into consideration, and a high-level representation is extracted by Higher-order Graph Neural Networks. On top of a sequence of UVGs, an attention network is built which can help to learn users' long-term voucher redemption preference. Extensive experiments on three large-scale production datasets demonstrate the proposed DMBGN model is effective, with 10% to 16% relative AUC improvement over Deep Neural Networks (DNN), and 2% to 4% AUC improvement over Deep Interest Network (DIN). Source code and a sample dataset are made publicly available to facilitate future research.