The ninth seed had eased past another American, Steve Johnson, in the third round and he built on that with another excellent performance on Court One, beating 30th seed Paul 6-4 7-5 6-4.
Having never previously been past the third round at a slam, Norrie is thriving on his first deep run and will be favoured to go further, with unseeded Belgian David Goffin his opponent in the last eight.
Norrie is known for his consistency and relentless athleticism but his forehand was the key weapon here from the moment he drilled a pass down the line on the first point.
Paul, also looking to reach his first slam quarter-final, will probably have nightmares about that shot, with Norrie using it time and again to take control in rallies.
Paul was facing a fourth consecutive left-handed opponent and had chosen Norrie as his practice partner to prepare for the previous three, so the 26-year-old’s game would not have come as a surprise.
Finding an answer, though, was another matter. Paul forced four break points in the sixth game but each time Norrie was rock solid, and that proved to be the American’s only chance to draw level in the opening set.
The second set followed a similar pattern until Norrie displayed his first signs of tension, failing to serve it out at 5-4.
It was Paul’s chance to really put his opponent under pressure but instead he played his worst service game of the match to give Norrie another opportunity, and this time he made no mistake.
The sort of tension in the crowd that often accompanies a British player at Wimbledon was absent, with Norrie appearing in complete control, and another early break in the third was enough to send him through to the last eight in style.