We investigate fast-pulse switching of in-plane-magnetized magnetic tunnel junctions (MTJs) within 3-terminal devices in which spin-transfer torque is applied to the MTJ by the giant spin Hall effect. We measure reliable switching, with write error rates down to 10-5, using current pulses as short as just 2 ns in duration. This represents the fastest reliable switching reported to date for any spin-torque-driven magnetic memory geometry and corresponds to a characteristic time scale that is significantly shorter than predicted possible within a macrospin model for in-plane MTJs subject to thermal fluctuations at room temperature. Using micromagnetic simulations, we show that in the three-terminal spin-Hall devices the Oersted magnetic field generated by the pulse current strongly modifies the magnetic dynamics excited by the spin-Hall torque, enabling this unanticipated performance improvement. Our results suggest that in-plane MTJs controlled by Oersted-field-assisted spin-Hall torque are a promising candidate for both cache memory applications requiring high speed and for cryogenic memories requiring low write energies.