Sutton resigned in April following Varnish's allegations that he told her to "go and have a baby" after she was dropped from the programme the previous month.
He has always denied the allegations.
"Following an internal investigation, the British Cycling board has upheld an allegation made by Jess Varnish that former Technical Director Shane Sutton had used inappropriate and discriminatory language," a British Cycling statement said.
"The board wishes to put on record its sincere regret that this happened. The findings of the investigation will help the development of the organisation alongside the independent review into the culture of the world class programme, jointly commissioned by British Cycling and UK Sport, and led by Annamarie Phelps."
It said the full text of the board's report had been made available to the independent review.
Varnish welcomed the findings of the investigation.
"I spoke out because I wanted to shine a light on the culture at British Cycling, a culture that in my mind was incorrect," the 25-year-old said in a statement.
"I’m relieved that the British Cycling board have acknowledged that the language used towards me was inappropriate and discriminatory.
"So I hope that British Cycling can use this investigation as a way to improve and create a better environment for the Great Britain team."
Sutton's resignation came about amid wider accusations of a culture of bullying and discrimination in British Cycling, claims supported by former Olympic champions Nicole Cook and Victoria Pendleton. Allegations also surfaced that he called Paralympic cyclists "wobblies".
Australia-born Sutton was a key component in Britain's ride to become a cycling superpower, first joining the organisation in 2002. He took over the role of technical director in 2014 when Dave Brailsford left to concentrate on running Team Sky.
Despite a third Tour de France win for Chris Froome and domination of the track cycling podium at the Rio Olympics, British Cycling has endured a torrid year.
It has been dragged into the controversy surrounding Team Sky's use of Therapeutic Use Exemptions (TUEs), including for former Tour de France winner Bradley Wiggins, and is part of a UK Anti-Doping investigation into allegations of "wrongdoing".
Last week chief executive of eight years Ian Drake said he would stand down in April.