Films made of 2D networks of single-walled carbon nanotubes (SWNTs) are one of the most promising active-channel materials for field-effect transistors (FETs) and have a variety of flexible electronic applications, ranging from biological and chemical sensors to high-speed switching devices. Challenges, however, still remain due to the current hysteresis of SWNT-containing FETs, which has hindered further development. A new and robust method to control the current hysteresis of a SWNT-network FET is presented, which involves the non-volatile polarization of a ferroelectric poly(vinylidene fluoride-trifluoroethylene) (P(VDF-TrFE)) gate insulator. A top-gate FET with a solution-processed SWNT-network exhibits significant suppression of the hysteresis when the gate-voltage sweep is greater than the coercive field of the ferroelectric polymer layer (≈50 MV m−1). These near-hysteresis-free characteristics are believed to be due to the characteristic hysteresis of the P(VDF-TrFE), resulting from its non-volatile polarization, which makes effective compensation for the current hysteresis of the SWNT-network FETs. The onset voltage for hysteresis-minimized operation is able to be tuned simply by controlling the thickness of the ferroelectric film, which opens the possibility of operating hysteresis-free devices with gate voltages down to a few volts.