Powell, who was appointed to the post in September 2015, is one of 20 Labour shadow cabinet members to quit in the wake of a referendum that saw 51.9 per cent of voters call for the UK to leave the European Union.
A total of 18 shadow junior ministers have also resigned, as Corbyn faces increasing pressure to resign amid criticism of his role in the "remain" campaign. Among them is shadow children's minister, Sharon Hodgson.
Corbyn has said that he intends to stay on as Labour leader and has appointed a swathe of replacements, including Pat Glass, MP for Durham North West, as shadow education secretary.
Glass was previously appointed shadow childcare minister by Corbyn in September 2015, but became shadow Europe minister in a reshuffle in January this year.
In her resignation letter, Powell described Corbyn's position as "untenable".
"You are unable to command the support of the shadow cabinet, the parliamentary Labour party and, most importantly, the country," the letter states.
"Given the big challenges faced by our country and our party over the coming months, and the possibility of an imminent general election, I do not have the confidence that you can bring the country together, lead us to a general election and be an effective opposition."
Among the other resignations is Gloria De Piero, shadow minister for young people and voter registration. She has been replaced by Cat Smith, MP for Lancaster and Fleetwood.
Shadow minister for education, Jenny Chapman, has also quit her position in the Labour education team. She said the party needs a leader who can "reach out more widely".
Jess Phillips, MP for Yardley and parliamentary private secretary in the shadow education team, and Lisa Nandy, shadow energy secretary and former policy director of The Children's Society, have also announced their resignations, as well as Luciana Berger, shadow cabinet minister for mental health.
Neil Leitch, chief executive of the Pre-school Learning Alliance, welcomed the appointment of Pat Glass as shadow education secretary.
"With a strong background in education and having only recently left her position as shadow childcare minister, she is particularly well placed to direct and inform Labour policy on early education and care," he said.
"At a time of exceptional political uncertainty, we look forward to working with Ms Glass to monitor the progress of childcare policy in England, and to ensure that the early years sector, and the sustainability of the service we deliver to children and families, is adequately supported going forward."