Norrie, the British No. 1, said he was jetlagged after defeating Japan's Kaichi Uchida 6-2, 6-2 in the previous round.
Brooksby will face Denis Shapovalov for a place in the final.
Shapavalov defeated Radu Albot in straight sets to advance to the semifinals.
The Canadian broke Albot twice in both sets to secure 6-2, 6-2 victory and book his spot in the last four.
Ruud, the world No. 2, brought the match level at 1-1 but Japan's Nishioka clinched the third set to set up a semifinal tie with American Aleksandar Kovacevic, who came back to down compatriot Mackenzie McDonald 4-6, 6-3, 6-2.
Information from the Associated Press contributed to this report.