Norrie continued his good form at his home Grand Slam, beating American Paul in straight sets in 2 hours, 24 minutes.
Norrie, Britain's No. 1, beat Paul with his first match point after being pushed hard in each set, with his expansive, left-handed forehand causing issues for the American.
Paul, seeded 30th, had defeated left-handed players in each of his previous rounds in straight sets. He was well-prepared for the grass major after playing on the surface in Hertogenbosch, Queen's Club Championships and Eastbourne.
Niemeier, 22, had her serve broken by Watson just once in the match, and cruised to straight-set victory in 1 hour, 18 minutes.
"I think in the women's game, often people are breaking through later," Watson said after the match.
"I think experience counts for a lot. Everybody's journey is different. This is mine. I broke a barrier this week. So I am definitely hungry for more."
Information from Reuters was used in this report.