Bag-of-Tasks (BoT) scheduling over cloud computing resources called Cloud Bag-of-Tasks Scheduling (CBS) problem, which is a well-known NP-hard optimization problem. Whale Optimization Algorithm (WOA) is an effective method for CBS problems, which still requires further improvement in exploration ability, solution diversity, convergence speed, and ensuring adequate exploration–exploitation tradeoff to produce superior scheduling solutions. In order to remove WOA limitations, a hybrid oppositional differential evolution-enabled WOA (called h-DEWOA) approach is introduced to tackle CBS problems to minimize workload makespan and energy consumption. The proposed h-DEWOA incorporates chaotic maps, opposition-based learning (OBL), differential evolution (DE), and a fitness-based balancing mechanism into the standard WOA method, resulting in enhanced exploration, faster convergence, and adequate exploration–exploitation tradeoff throughout the algorithm execution. Besides this, an efficient allocation heuristic is added to the h-DEWOA method to improve resource assignment. CEA-Curie and HPC2N real cloud workloads are used for performance evaluation of scheduling algorithms using the CloudSim simulator. Two series of experiments have been conducted for performance comparison: one with WOA-based heuristics and another with non-WOA-based metaheuristics. Experimental results of the first series of experiments reveal that the h-DEWOA approach results in makespan improvement in the range of 5.79–13.38% (for CEA-Curie workloads), 5.03–13.80% (for HPC2N workloads), and energy consumption in the range of 3.21–14.70% (for CEA-Curie workloads) and 10.84–19.30% (for HPC2N workloads) over well-known WOA-based metaheuristics. Similarly, h-DEWOA also resulted in significant performance in comparison with recent state-of-the-art non-WOA-based metaheuristics in the second series of experiments. Statistical tests and box plots also revealed the robustness of the proposed h-DEWOA algorithm.