Traditional decision-based black-box adversarial attacks on image classifiers aim to generate adversarial examples by slightly modifying input images while keeping the number of queries low, where each query involves sending an input to the model and observing its output. Most existing methods assume that all queries have equal cost. However, in practice, queries may incur asymmetric costs; for example, in content moderation systems, certain output classes may trigger additional review, enforcement, or penalties, making them more costly than others. While prior work has considered such asymmetric cost settings, effective algorithms for this scenario remain underdeveloped. In this paper, we propose a general framework for decision-based attacks under asymmetric query costs, which we refer to as asymmetric black-box attacks. We modify two core components of existing attacks: the search strategy and the gradient estimation process. Specifically, we propose Asymmetric Search (AS), a more conservative variant of binary search that reduces reliance on high-cost queries, and Asymmetric Gradient Estimation (AGREST), which shifts the sampling distribution to favor low-cost queries. We design efficient algorithms that minimize total attack cost by balancing different query types, in contrast to earlier methods such as stealthy attacks that focus only on limiting expensive (high-cost) queries. Our method can be integrated into a range of existing black-box attacks with minimal changes. We perform both theoretical analysis and empirical evaluation on standard image classification benchmarks. Across various cost regimes, our method consistently achieves lower total query cost and smaller perturbations than existing approaches, with improvements of up to 40% in some settings.