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Recent Deep Learning (DL) advancements in solving complex real-world tasks have led to its widespread adoption in practical applications. However, this opportunity comes with significant underlying risks, as many of these models rely on privacy-sensitive data for training in a variety of applications, making them an overly-exposed threat surface for privacy violations. Furthermore, the widespread use of cloud-based Machine-Learning-as-a-Service (MLaaS) for its robust infrastructure support has broadened the threat surface to include a variety of remote side-channel attacks. In this paper, we first identify and report a novel data-dependent timing side-channel leakage (termed Class Leakage) in DL implementations originating from non-constant time branching operation in a widely used DL framework PyTorch. We further demonstrate a practical inference-time attack where an adversary with user privilege and hard-label black-box access to an MLaaS can exploit Class Leakage to compromise the privacy of MLaaS users. DL models are vulnerable to Membership Inference Attack (MIA), where an adversary's objective is to deduce whether any particular data has been used while training the model. In this paper, as a separate case study, we demonstrate that a DL model secured with differential privacy (a popular countermeasure against MIA) is still vulnerable to MIA against an adversary exploiting Class Leakage. We develop an easy-to-implement countermeasure by making a constant-time branching operation that alleviates the Class Leakage and also aids in mitigating MIA. We have chosen two standard benchmarking image classification datasets, CIFAR-10 and CIFAR-100 to train five state-of-the-art pre-trained DL models, over two different computing environments having Intel Xeon and Intel i7 processors to validate our approach.