Laidlaw's four second-half penalties proved enough for the Scots, who had trailed 8-0 following debutant Paddy Jackson's first-half penalty and winger Craig Gilroy's try three minutes after the interval.
Ireland, who dominated possession throughout the match, camped near the Scottish line in the closing stages knowing a try would put them 13-12 up but the home side stayed strong to keep their frustrated visitors out.
Scotland, who last won the competition in 1999 when it was still the Five Nations and whose last successive wins came in the foot and mouth disease-affected 2001 tournament, are third in the table with four points.
They sit behind England on six and level with holders Wales, their next opponents on March 9 in what will be a key encounter given the pace-setters will be heavy favourites to notch up a fourth successive win when they host Italy.
"We are in it up to our ears and that's great for the fans and the team. But let's not get carried away," interim Scotland coach Scott Johnson told the BBC.
IRISH ONSLAUGHT
What most impressed Johnson was his side's grit and determination in the face of an Irish onslaught for the vast majority of an error-strewn game.
"I'm happy for the boys because it showed a different side to us - we showed great resolve. We could have been well behind at halftime," said Johnson whose team had 29 per cent of the possession.
"If you can combine that resolve with the skill sets we have then we can do some damage. I'm happy but there is still plenty we can improve on."
Scotland were buoyant coming into the game following their resounding win over Italy but they were soon on the back foot against Ireland who mounted wave after wave of attacks.
All one of the pre-tournament favourites had to show for their territorial dominance at halftime, however, was a three-point lead, much to the dismay of coach Declan Kidney.
"It's all about taking your opportunities. We created several try-scoring opportunities but we didn't convert them to take us in front of Scotland's penalty count," said Kidney, upset at several missed penalty attempts by Jackson.
"Their place-kicking stats were higher than ours and there will be some regret about that. We created opportunities, but I will take a good look at myself this week, as will the rest of the guys around me."
Gilroy scored the only try when he rounded off a spell of sustained Irish pressure by wriggling over the line from close range.
Jackson, 21, was brought in to replace injured regular flyhalf Jonathan Sexton, leaving Ireland's most-capped player Ronan O'Gara on the bench once again.
Ireland next face bottom team France at home on March 9 before they travel to Italy to round off a disappointing campaign that started brightly with a superb win over Wales in Cardiff.
