The indie-pop band's An Awesome Wave was the bookies' favourite for the award, and triumphed over albums from the likes of Plan B, Jessie Ware, The Maccabees, Field Music and Django Django.
Vocalist Joe Newman told Digital Spy: "It was definitely a pressure for us because there are so many strong contenders this year. We just felt lucky to be amongst such good quality artists really."
He added that the nomination alone had noticeably changed things for the band, saying: "We've noticed a change in the atmosphere at gigs. Everyone is a lot more vocal and shouty in a good way. There's more of a presence from the audience which I think is a result of the Mercury Prize."
Guitarist Gwil Sainsbury agreed, joking: "Our riders are better now as well, we get more respect!"
Photo gallery - Mercury Prize 2012 nominees arrive:
Copyright: PA Images Yui Mok/PA Wire
Sainsbury said of the win: "There was a real lack of dance on this year's list. This list is what it is - it's been chosen by individuals, which is always nice.
"Better than a computer that makes sure a bit of everything is in there. It's a panel of judges who love 12 albums, which is what makes it special. "
The Leeds band then headed off to celebrate their win at an after-party in Camden, adding: "We were going to celebrate... win or lose!"
Watch the video for Alt-J's 'Breezeblocks' below: