This paper proposes a compressed sensing-based random access protocol (CS-RACH), which is suitable for servicing a large number of machine-type communication devices in Internet of Things (IoT) network. In CS-RACH, we utilize a larger number of unique preambles compared to conventional LTE-RACH, however, the compressed sensing technique makes it possible to simultaneously detect the users with high accuracy. Compared to the user detection in conventional LTE-RACH, the proposed user detection can get rid of preamble collisions and decrease the collision probability, thereby the overall access latency is significantly reduced. To prove the benefits of the proposed CS-RACH, we mathematically analyze and compare access latency performance of LTE-RACH and CS-RACH. In particular, based on the least absolute shrinkage and selection operator approach, we derive a normalized throughput, access success probability, and average access latency. Our simulation results also exhibit that the proposed CS-RACH considerably reduces the access latency under reasonable conditions in IoT environments.