The band members, who met at Leeds University in 2007, looked stunned as they collected the award, despite long being the bookies' favourites.
After five years of hard work, the band beat acts such as rapper Plan B to win the award in London.
Accepting the title on stage, Alt-J – Thom Green (drums), Joe Newman (guitar/vocals), Gwil Sainsbury (guitarist/bassist) and Gus Unger-Hamilton (keyboards) – said there were too many people to thank, including their parents for "not making us get jobs".
Unger-Hamilton said the £20,000 would not change their lives too much. He said: "It won't nearly pay off our student loans."