After being pushed to three sets in his first two matches, the 11-time champion was sharp against the Brit, firing off 22 winners to Norrie's nine and breaking four times.
11-time champion plays best match of the week to reach the semifinals
Nadal was putting pressure on early, extending Norrie to deuce in his first service game and the next time the Brit served, he sprayed a forehand long to give the top seed a 3-1 advantage.
A 14-point game culminated in a second break after another miss on the forehand side by Norrie and a game later, he served out the set to love when the world number 58 misfired on yet another forehand.
The momentum stayed with Nadal and an incredible backhand winner landing in the corner on the stretch gave the Spaniard an early break in the second set and a 2-1 lead.
Norrie would keep pace and break back for 4-4, but more forehand errors gave Nadal a break to love. Serving for the match at 5-4, he needed two match points, a big serve sealing the victory on his second chance.