This paper presents an energy-efficient hardware accelerator for binarized convolutional neural networks (BCNNs). In this BCNN accelerator, a data-shift operation becomes dominant to effectively control input/weight-data streams under limited memory bandwidth. A magnetic-tunnel-junction (MTJ)-based nonvolatile field-programmable gate array (NV-FPGA), where the amount of stored-data updating is minimized in a configurable logic block, is a well-suited hardware platform for implementing such a BCNN accelerator. Owing to the nonvolatile storage capability of the NV-FPGA, not only power consumption in the data-shift operation but also standby power consumption in the idle function block is reduced without losing internal data. It is demonstrated under 45 nm complementary metal-oxide-semiconductor/MTJ process technologies that the energy consumption of the proposed BCNN accelerator is 50.7% lower than that of a BCNN accelerator using a conventional static-random-access-memory-based FPGA.