Laidlaw landed all four of his penalty attempts while, in contrast, debutant Ireland outside-half Paddy Jackson managed just one successful goal-kick from four attempts.
Scotland, under the interim charge of Australian coach Scott Johnson, were 8-0 down early in the second half after Ireland wing Craig Gilroy crossed for the only try of the match.
But Scotland's forwards, dominant at the scrum, eventually laid a points-scoring platform for Laidlaw, who took his side into a 12-8 lead with barely five minutes left.
Ireland then pressed for a match-winning try but when debutant centre Luke Marshall knocked on with the final play, the Scots, roared on by a crowd of over 67,000, had done enough to secure victory in a match where the visitors had 80 percent possession.
"Credit to all the forwards, we really dug deep today, and I think you can see that means a lot to us," said Scottish lock and man-of-the-match Jim Hamilton. "The scrum was immense.
"We knew Ireland notoriously start very well, we hung in there in the first half and to go in 3-0 to Ireland was probably a good outing for us. We had to come out in the second half, we dug deep and took our chances."
Johnson added: "That was a game we probably shouldn't have won but we did and it showed great spirit and great heart.
"We're still in the tournament, up to our ears, and that's great."
Ireland counterpart Declan Kidney was left ruing his side's inability to notch up more points.
"Their place-kicking went at a far higher stat than us," he said. "We did do a lot of positive work in attack but we didn't manage to finish it off.
"We gave away a few penalties and we'll take a deep look at ourselves."
Scotland made just one change to the side that beat Italy 34-10, with prop Geoff Cross replacing Euan Murray, who doesn't play on Sundays because of his Christian beliefs.
Injuries and suspension forced Ireland into making five changes from the side that lost 12-6 to England last time out, with Kidney taking the bold decision to hand a Test debut to 21-year-old Ulsterman Paddy Jackson rather than recall veteran Ronan O'Gara in place of hamstring victim Jonathan Sexton.
Jackson's Ulster team-mate and centre Marshall also made his Test debut, and the duo made a neat scissors move in midfield, with the latter drawing Scotland full-back Stuart Hogg only to then produce a forward pass out to Gilroy when the wing was clear on the right touchline.
The tournament resumes on the weekend of March 9-10 when Scotland entertain Wales, Ireland play France and unbeaten England welcome Italy.