Streaming videos over cellular networks is highly challenging. Since cellular data is a relatively scarce resource, many video and network providers offer options for users to exercise control over the amount of data consumed by video streaming. Our study shows that existing data saving practices for Adaptive Bitrate (ABR) videos are suboptimal: they often lead to highly variable video quality and do not make the most effective use of the network bandwidth. We identify underlying causes for this and propose two novel approaches to achieve better tradeoffs between video quality and data usage. The first approach is Chunk-Based Filtering (CBF), which can be retrofitted to any existing ABR scheme. The second approach is QUality-Aware Data-efficient streaming (QUAD), a holistic rate adaptation algorithm that is designed ground up. We implement and integrate our solutions into two video player platforms (dash.js and ExoPlayer), and conduct thorough evaluations over emulated/commercial cellular networks using real videos. Our evaluations demonstrate that compared to the state of the art, the two proposed schemes achieve consistent video quality that is much closer to the user-specified target, lead to far more efficient data usage, and incur lower stalls.