Electrostatic gating of two-dimensional (2D) materials with ionic liquids (ILs), leading to the accumulation of high surface charge carrier densities, has been often exploited in 2D devices. However, the intrinsic liquid nature of ILs, their sensitivity to humidity, and the stress induced in frozen liquids inhibit ILs from constituting an ideal platform for electrostatic gating. Here we report a lithium-ion solid electrolyte substrate, demonstrating its application in high-performance back-gated n-type MoS2 and p-type WSe2 transistors with sub-threshold values approaching the ideal limit of 60 mV/dec and complementary inverter amplifier gain of 34, the highest among comparable amplifiers. Remarkably, these outstanding values were obtained under 1 V power supply. Microscopic studies of the transistor channel using microwave impedance microscopy reveal a homogeneous channel formation, indicative of a smooth interface between the TMD and underlying electrolytic substrate. These results establish lithium-ion substrates as a promising alternative to ILs for advanced thin-film devices.