In a statement issued by the governing body on Wednesday, Sutton, who has denied all the allegations, said his departure was in the best interests in British Cycling.
"The developments over the past few days have clearly become a distraction. It is for this reason that I believe it is in the best interests of British Cycling for me to step down from my position as technical director," he said.
"It is important that the review announced by British Cycling and UK Sport now takes place, and I will obviously co-operate fully with this.
The claims have left the sport in turmoil 100 days before it is expected to put up a strong showing at the Olympic Games in Rio.
They first surfaced in a newspaper interview by Varnish in the Daily Mail. The former European sprint champion said Sutton told her to "move on and have a baby" and that her "ass" was too big.
British Olympic cyclists Victoria Pendleton and Nicole Cooke backed her claims of sexism in British Cycling.
On Wednesday, a new row blew up over Paralympians when Darren Kenny, one of Britain’s most decorated para-cyclists, told the Daily Mail: “The term used to refer to us was generally ‘gimps’ with another word in front of that.
"I don’t think we were considered to be elite athletes in all honesty by certain people and since I’ve left I’ve not seen or heard of any change towards equality.”
The paper also quoted a source saying: “They were referred to as wobblies. It was an in-joke used to describe para-cyclists.”
In Wednesday's statement, Sutton denied the allegations made against him.
"I have made clear that I reject the specific claims that have been made against me in recent days, and I look forward to taking a full part in the review process so I can respond to the allegations in detail."
His departure is a blow to British Cycling. The 58-year-old, who joined the organisation in 2002, played a major part in the Olympic success of Chris Hoy and Bradley Wiggins before taking over the top job from Dave Brailsford.
British Cycling chief executive Ian Drake thanked Sutton for his work with British Cycling and "the part he has played in our success.
"I understand and respect Shane's decision to stand down. His primary focus has always been the athletes, and this decision is something he has taken to allow them to focus on their preparation for Rio," said Drake.
He added that programmes director Andy Harrison would take over Sutton's role.