We report the discovery of NGTS-11 b (=TOI-1847b), a transiting Saturn in a 35.46 day orbit around a mid K-type star (T_eff_=5050+/-80K). We initially identified the system from a single-transit event in a TESS full-frame image light curve. Following 79 nights of photometric monitoring with an NGTS telescope, we observed a second full transit of NGTS-11 b approximately one year after the TESS single-transit event. The NGTS transit confirmed the parameters of the transit signal and restricted the orbital period to a set of 13 discrete periods. We combined our transit detections with precise radial-velocity measurements to determine the true orbital period and measure the mass of the planet. We find NGTS-11 b has a radius of 0.817_-0.032_^+0.028^R_Jup_, a mass of 0.344_-0.073_^+0.092^M_Jup_, and an equilibrium temperature of just 435_-32_^+34^K, making it one of the coolest known transiting gas giants. NGTS-11 b is the first exoplanet to be discovered after being initially identified as a TESS single-transit event, and its discovery highlights the power of intense photometric monitoring in recovering longer-period transiting exoplanets from single-transit events.