In this paper, we provide a simple framework for deriving one-shot achievable bounds for some problems in quantum information theory. Our framework is based on the joint convexity of the exponential of the collision relative entropy, and is a (partial) quantum generalization of the technique of Yassaee et al. (2013) from classical information theory. Based on this framework, we derive one-shot achievable bounds for the problems of communication over classical-quantum channels, quantum hypothesis testing, and classical data compression with quantum side information. We argue that our one-shot achievable bounds are strong enough to give the asymptotic achievable rates of these problems even up to the second order.