Field-effect transistor (FET) biosensors based on two-dimensional (2D) materials have drawn significant attention due to their outstanding sensitivity. However, the Boltzmann distribution of electrons imposes a physical limit on the subthreshold swing (SS), and a 2D-material biosensor with sub-60 mV/dec SS has not been realized, which hinders further increase of the sensitivity of 2D-material FET biosensors. Here, we report tunnel FETs (TFETs) based on a SnSe2/WSe2 heterostructure and observe the tunneling effect of a 2D material in aqueous solution for the first time with an ultralow SS of 29 mV/dec. A bilayer dielectric (Al2O3/HfO2) and graphene contacts, which significantly reduce the leakage current in solution and contact resistance, respectively, are crucial to the realization of the tunneling effect in solution. Then, we propose a novel biosensing method by using tunneling current as the sensing signal. The TFETs show an extremely high pH sensitivity of 895/pH due to ultralow SS, surpassing the sensitivity of FET biosensors based on a single 2D material (WSe2) by 8-fold. Specific detection of glucose is realized, and the biosensors show a superb sensitivity (3158 A/A for 5 mM), wide sensing range (from 10-9 to 10-3 M), low detection limit (10-9 M), and rapid response rate (11 s). The sensors also exhibit the ability of monitoring glucose in complex biofluid (sweat). This work provides a platform for ultrasensitive biosensing. The discovery of the tunneling effect of 2D materials in aqueous solution may stimulate further fundamental research and potential applications.