The pair were handed the £20,000 prize at a ceremony at Glasgow’s Film City complex for their joint album “Everything’s Getting Older.”
The new award, instigated by the Scottish Music Industry Association, had been given £100,000 in funding from Creative Scotland to get off the ground.
Wells and Moffat’s album of “jazz-inflected love songs”, which they said had been almost 10 years in the making, fought off challenges from the likes of King Creosote and Jon Hopkins, Tommy Smith, Twin Atlantic and Mungo’s Hi Fi.
Moffat said: “I really didn’t expect us to win.
“Obviously we stood a one in ten chance but I was convinced Mogwai were going to win.
“I couldn’t have imagine something like this when we made the album but you don’t make a record to win awards and this event didn’t exist before.
“I’m not sure why nobody thought of having this before. Scotland has always produced a very high quality of music and there should be something that reflects that in a way that other awards don’t. There’s just not enough room anywhere else.
“I’m not sure what we’ll do with the money but it’s certainly a lot for musicians to get these days. It’s very difficult to make money.”
Moffat, a founding member of Arab Strap, said it was the biggest prize he had collected since winning a watch at fast food restaurant Wimpy when he was a child.
Scottish culture secretary Fiona Hyslop, who gave out the award, said: “This new award is a fantastic celebration of Scottish contemporary music talent which enhances our international reputation as a hugely talented and creative nation.”
The 10 short-listed acts were all given a £1,000 runner-up prize, as well as a piece of artwork designed by Glasgow School of Art student Fraser Clark.
The award was judged by a panel made up of 100 music and arts industry figures, including writers for The Scotsman and Scotland on Sunday.