The notion of diagnosability has long played an important role in measuring the reliability of multiprocessor systems. Such a system is $t$-diagnosable if all faulty nodes can be identified without replacement when the number of faults does not exceed $t$, where $t$ is some positive integer. Furthermore, a system is strongly $t$-diagnosable if it can achieve $(t+1)$-diagnosability, except for the case where a node's neighbors are all faulty. In this paper, we investigate the strong diagnosability of a class of product networks, under the comparison diagnosis model. Based on our results, we can determine the strong diagnosability of several widely used multiprocessor systems, such as hypercubes, mesh-connected $k$-ary $n$-cubes, torus-connected $k$-ary $n$-cubes, and hyper Petersen networks.