Random access (RA) scheme is one of the critical issues for successfully and efficiently connecting devices in the Internet of Things (IoT) networks. In cellular-based IoT networks, Zadoff-Chu (ZC) sequences are used as preambles, where their cyclically shifted versions are orthogonal to each other. We propose a time-shifted RA scheme for cellular IoT networks to reduce collision and to accommodate more devices, which can also increase fairness. In the proposed scheme, each IoT device efficiently adjusts the amount of cyclic shift of ZC sequences during preamble generation phase. We formulate an optimization problem to minimize average collision probability and provide an algorithmic solution. Simulation reveals that our proposed scheme can achieve near-perfect fairness while effectively mitigating collision probability.