The 23-year-old Leinster all-rounder, who has won 78 Irish caps, takes over after Joyce quit in the wake of the group-stage exit in the World Twenty20.
"It's a really proud moment for me and a really exciting time in my playing career," Delany said.
"It'll be a privilege to captain this side. As a squad we know our potential and what we are capable of achieving."
She added: "For a captain, that is incredibly exciting and we can't wait for what the summer has in store.
"It shows the belief Cricket Ireland, the selectors and coaching staff have in me both as a player and person.
"Certainly it is a big responsibility and the standards that have been set before me are extremely high."
Ireland women's head coach Aaron Hamilton explained the rationale in selecting Delany for the role.
"She has a maturity beyond her years, with an old head on young shoulders," he said.
"Laura is very astute, possessing a shrewd cricketing brain who reads the game well, and is cool-headed.
"She is an inspirational leader who manages relationships well and is widely respected by her peers and team management."
Cricket Ireland also announced that 19-year-old Pembroke all-rounder Kim Garth will be the new vice-captain.
Laura Delany will succeed Isobel Joyce as the new captain of the Ireland women's team.