Britain's Norrie beats Paul to reach Wimbledon quarters
Norrie was quickly into his stride on Court One, breaking the American 30th seed at his first opportunity and saving four break points in a marathon sixth game.
He served out to love to take the first set and again broke early in the second only to falter when serving for a two-set lead.
But the South Africa-born left-hander broke again immediately and this time made no mistake on his serve.
A single break in the third game of the third set proved decisive as the 26-year-old sealed a place in the last eight against Belgium's David Goffin.