Novak Djokovic, Rafael Nadal, Roger Federer, Naomi Osaka and Serena Williams were among those skipping the tournament that was delayed 2 1/2 years because of the coronavirus.
Angelique Kerber, the 2019 runner-up and highest remaining seed at No.10, lost to 21st-seeded Paula Badosa 6-4, 7-5.
Rublev, ranked No.5 in the world, claimed the first set with a scorching 117 mph ace before Norrie, ranked No. 28, rebounded to force a decisive third set to reach his fifth final of the year.
Wimbledon: Roger Federer bids to silence Cameron Norrie's home support, Ashleigh Barty seeks to fire at last
Federer's clinical performance against Richard Gasquet on Thursday contrasted to an error-strewn one by Australia's women's world number one Ashleigh Barty.