The former world number one extended his winning streak to six matches after successfully defending his title at last week's Almaty Open.
After dropping the opening set, the third seed fought back to force a decider and twice recovered from a break down in the third, saving a match point when Norrie served for victory before dominating the tiebreak 7-1.
Medvedev's frustrations spilled over early in the final set when he repeatedly bent down and struck the court after falling 0-2 behind, but the Russian regrouped to complete the win and keep alive his chances of securing a spot at the season-ending ATP Finals.
Medvedev will next face French qualifier Adrian Mannarino, who stunned eighth seed Jakub Mensik in the round of 16.
Carlos Alcaraz also later earned his 60th tour-level quarter-final appearance with a 6-1, 7-6(11) victory over Nikoloz Basilashvili in the second round of the Vienna Open, saving four set points in a dramatic second-set tiebreak.