After being broken in his first service game of the match and going on to lose the first set, Norrie went a break up early in the second.
However, Ruud hit back to make it 3-3 and then broke to love to clinch the victory.
Italian top seed Jannik Sinner remains on course for his fourth title of 2024 after he saw off Russia’s Pavel Kotov 6-2 7-5.
There were five consecutive breaks of serve in the first set before Sinner eventually held to go 5-2 up.
Another break secured the set but the Australian Open champion, who has only lost twice this year, was under pressure in the second.
Kotov had the chance to force a deciding set as he served at 5-4 up in the second but Sinner broke him twice in a row to advance.
Third seed Daniil Medvedev also progressed after coming from a set down to beat America’s Sebastian Korda 5-7 7-6 (7-4) 6-3.
