Light-flyweight Yafai, 23, scored a clear points victory over Cameroon's Simplice Fotsala in the round of 32.
Welsh lightweight Cordina used classy footwork and counterpunching to beat bustling Filipino Charly Suarez.
Okolie, a heavyweight in just his 26th bout, used his jab well to overcome Polish fighter Igor Jakubowski.
The 23-year-old Londoner only quit his job at McDonalds after watching Anthony Joshua win super-heavyweight gold at London 2012, and battled obesity as a youth.
"To become a legend, you have to do legendary stuff. This is just part of the journey to be a legend," he said.
"If I manage to win, then I deserve it. It's not about scraping a medal or getting lucky. It's about wanting to beat everyone. If I can beat everyone in front of me, I get gold. If I don't, I don't deserve it."
Light-heavyweight Joshua Buatsi is the only Briton in action on Sunday, taking on Uganda's Kennedy Katende at 17:00 BST.
British fighters Galal Yafai, Joe Cordina and Lawrence Okolie progressed in the Olympic boxing competition on the first day of Rio 2016.