They also won best British group and best British single for Drag Me Down.
The prize was collected on the band's behalf by DJ Greg James, who revealed that One Direction will play a special Live Lounge on Radio 1 on November 12.
Best film was given to "The Hunger Games: Mockingjay Part 1", while "The X Factor" picked up best British TV show.
The ceremony also celebrated the achievements of three Teen Heroes.
The 2015 Teen Heroes are Alice, a 14-year-old carer from Gloucestershire, Rosie, 14, who is a fundraiser from Coventry with a highly malignant primary brain tumour and Aaren, a 16-year-old from Bristol who was born with gender dysphoria.