Ellen White and Jayne Ludlow fired the Women's Super League and FA Cup winners into a first-half lead.
Yankey struck twice either side of a Isobel Christiansen goal for Birmingham, who had striker Rachel Williams sent off in stoppage time.
"To win in style was a massive positive," said Gunners boss Laura Harvey.
"We knew it was going to be tough against a Birmingham side who are always looking to get the ball into your penalty area.
"We started the game badly, but we defended well throughout and the only real hiccup was their goal."