The scouts from Sheffield and Doncaster rushed to help after hearing the stricken walker's cries while they were taking part in a challenge on Kinder Scout.
Using their training, the teenagers from Woodseats Venture Unit and Tickhill-based Phoenix Explorer Scouts gave first aid, kept him warm and raised the alarm.
The walker, a 19-year-old Sheffield Hallam University student who has not been named and was not involved in the scouts' event, is recovering in the Northern General Hospital following the incident on Saturday.
Both bones in his lower right leg had been broken.
Ben Nuttall, aged 17, from the Woodseats unit, said he and team mates Andrew Batley, 17, Andrew Taylor, 17, and Louis Sheeron, 16, were on the last stage of the trek when they heard shouting.
He said: "We saw the man lying in a gully with his friend and were told he had a broken leg. Andrew Taylor and I are trained lifeguards and we all have general first aid qualifications, so we knew what to do.
"We sorted him out, got him into some warm clothes and put a survival blanket round him and checked his leg out. His ankle was quite bruised, so we carefully removed his shoe and sock, and gave him another sock.
"He was in shock, and shaking quite badly, so we ensured he was warm enough and gave him some chocolate to ensure he had sugar inside him."
They were joined at the scene in Grindsbrook Clough, on the way down from the summit of Kinder Scout to Edale, by three scouts from Phoenix group, Doncaster, who were following behind.
Jo Taylor, 16, stayed with the Sheffield team, while Mark Harrop, 16, and Adam Walker, 16, went to find help.
Mark said: "We walked back up to the plateau at the top of the clough and found a phone signal, so we rang the organisers, who called Mountain Rescue."
The teams discovered the man at about 4pm, and although Edale Mountain Rescue arrived within 30 minutes, it was not until 10.30pm that they managed to bring him down to safety.
James Stubley, deputy leader of Edale Mountain Rescue, said: "We had requested an RAF helicopter but it was windy and there was low cloud so it was unable to reach the scene.
"We had to carry the man down to Edale, where he was taken to the Northern General Hospital by ambulance."
Mr Stubley added: "Grinds-brook Clough is one of the main routes but at that time there are only limited numbers of people coming back down and there was a danger of hypothermia.
"It was fortunate those couple of groups of scouts heard him shouting, otherwise his rescue could have taken longer as his friend would have had to walk down to seek help.
"The scouts handled things very well, keeping him warm and giving an accurate location so we knew where to reach him."
The rescue happened during the Apex Challenge, entered by 150 scouts aged from 14 to 25 from across the UK, who took part in 40 teams.
They spent two days in the Peak District tackling tough challenges such as climbing, abseiling and canoeing, while also battling for points.
Event Organiser Matt Richards said: "This mountain rescue incident just summed up the spirit of the event. Despite the young people racing against the clock to complete their challenges, they didn't hesitate to stop and help this member of the public who clearly needed assistance.
"It's great that we were able to do our bit to help, and showed the value of the training they've been given during their time in scouting. These teenagers can be very proud of what they did."