Nowadays, increase in time complexity of applications and decrease in hardware costs are two major contributing drivers for the utilization of high-performance architectures such as cluster computing systems. Actually, cluster computing environments, in the contemporary sophisticated data centres, provide the main infrastructure to process various data, where the biomedical one is not an exception. Indeed, optimized task scheduling is key to achieve high performance in such computing environments. The most distractive assumption about the problem of task scheduling, made by the state-of-the-art approaches, is to assume the problem as a whole and try to enhance the overall performance, while the problem is actually consisted of two disparate-in-nature subproblems, that is, sequencing subproblem and assigning one, each of which needs some special considerations. In this paper, an efficient hybrid approach named ACO-CLA is proposed to solve task scheduling problem in the mesh-topology cluster computing environments. In the proposed approach, an enhanced ant colony optimization (ACO) is developed to solve the sequence subproblem, whereas a cellular learning automata (CLA) machine tackles the assigning subproblem. The utilization of background knowledge about the problem (i.e., tasks' priorities) has made the proposed approach very robust and efficient. A randomly generated data set consisting of 125 different random task graphs with various shape parameters, like the ones frequently encountered in the biomedicine, has been utilized for the evaluation of the proposed approach. The conducted comparison study clearly shows the efficiency and superiority of the proposed approach versus traditional counterparts in terms of the performance. From our first metric, that is, the NSL (normalized schedule length) point of view, the proposed ACO-CLA is 2.48% and 5.55% better than the ETF (earliest time first), which is the second-best approach, and the average performance of all other competing methods. On the other hand, from our second metric, that is, the speedup perspective, the proposed ACO-CLA is 2.66% and 5.15% better than the ETF (the second-best approach) and the average performance of all the other competitors.