Massaro, the fifth seed and 2013 champion, beat Egypt's third seed Raneem el Welily 11-3 13-15 11-6 11-6.
Seventh seed Perry joined Massaro with a dramatic 11-4 7-11 13-11 6-11 11-4 win over Australia's Donna Urquhart.
In the men's draw, two-time champion and fourth seed Matthew saw off Egyptian eighth seed Tarek Momen 11-9 10-12 11-6 7-11 11-6.
Matthew next faces top seed Mohamed Elshorbagy, who was in trouble when he trailed fellow Egyptian Ali Farag in the fourth game, but recovered to win 11-8 9-11 8-11 12-10 11-5.
Massaro, 33, goes on to face Egypt's top seed and reigning champion Nour el Sherbini in a repeat of of last year's World Championships final, which the Egyptian won from 0-2 down.
"It was a big quarter-final," Massaro told the PSA World Tour. "It always feels like a big match when you play Raneem, it definitely felt bigger than a quarter-final."
El Sherbini, 21, beat England's 13th seed Emily Whitlock 11-3 11-4 11-4 in the first of the quarter-finals.
England's Laura Massaro, Sarah-Jane Perry and Nick Matthew are through to the British Open semi-finals in Hull.