Glasgow's Dan Parks won his third man-of-the-match award of the tournament with an outstanding display that included 18 points and the winning penalty in 79th minute.
His Warriors team-mate Johnnie Beattie also scored a try for the visitors, while Brian O'Driscoll touched down for Ireland, whose other points were kicked by Jonathan Sexton and Ronan O'Gara as they missed out on a fifth triple crown in seven years.
"Obviously it feels fantastic. I think we've deserved a lot more than we've got in this championship.
"We came here full-blooded and we're absolutely delighted. It's going to be a good night now."
The winning kick was a daunting one and the jeers of the Irish crowd turned up the pressure but Dan remained focused, and said: "I was just delighted I got a good strike on it and it went through."
Scotland coach Andy Robinson said: "The guys have worked hard all year.
"We got a bit of luck at the end of the game, got a penalty near their line and Dan made a great kick, but I think for the endeavour and the way we've tried to play this season, the guys should take some real credit.
"They're hurting, they're tired, and they deserve the plaudits for this win.
"I'm impressed with the way they have approached this game and approached every game.
"Like all things we kept the composure right to the end.
"Ireland played really, really well at the start and I thought at one stage it was going to be a 30-pointer.
"But we held on, we got to the pace of the game, and performed really well."
Captain Chris Cusiter admitted the way Scotland dealt with Ireland drawing level at 17-17 proved they had improved from their cruel defeat in Wales five weeks earlier.
"That was the test for us," the scrum-half said.
"With what's happened this championship and how some of the other games have gone, I'm really pleased with what happened in that period.
"I knew that when they got back level with us, we had to earn the win.
"For me, that was a significant step from what we've been doing at the end of the games."