Laidlaw's dramatic last-gasp penalty kick secured them a narrow 29-26 win over Italy in Rome.
Italy had led 17-12 at half-time but suffered their 17th straight defeat after a 79th-minute penalty by Laidlaw handed Scotland victory.
"I'm destroyed for the lads and the fans but we've seen our future, the style and the game that we want, the quality of our young players," said Italy coach Conor O'Shea.
Scotland move provisionally second in the Six Nations table but a win for either France or Wales, who play each other in Cardiff later Saturday, will move them to third.
Italy finish bottom of the table for the third straight year with five defeats in five games, with captain Sergio Parisse suffering his 100th Test defeat.
Italy's losing streak in the Six Nations stretches back to 2015 against Scotland in Murrayfield.
The Italians had gotten off to a blistering start with Tommaso Allan, who played for Scotland at U-20 level, winning the man of the match award as he accounted for 22 of Italy's points.
The 24-year-old touched down after 14 and 45 minutes with Matteo Minozzi getting a third for the hosts after 21 minutes.
Scotland's Fraser Brown and John Barclay both scored tries, in the first half for Scotland, with Sean Maitland and Stuart Hogg adding two more in the second half.
Scotland led 5-3 after ten minutes when Brown touched down after Allan's opening penalty, but Laidlaw sent the conversion wide.
But Allan pulled Italy ahead 12-5 after quarter-of-an-hour to the delight of the 60,000 crowd with Minozzi slotting down as the hosts opened up a 17-5 lead after 22 minutes.
Back after the break, Sebastian Negri looked to have scored a try as he burst through the post to dot down but it was disallowed because of a knock-on in the build-up.
But two minutes later Allan broke as newcomer Jake Polledri created an opportunity from following a sloppy turnover in the Scottish midfield offloading to Allan who raced through and then converted to open up a 12-point lead after 46 minutes.
Maitland pushed through for another try with Laidlaw converting from a tight angle to pull Scotland 24-19, before Hogg pulled the Scots level and Laidlaw converting.
Allan edged Italy ahead again with a penalty but Laidlaw slotted in a penalty in the dying seconds for his ninth point of the afternoon to break Italian hearts.
Scotland's title hopes had been extinguished with defeat to Ireland last week, as Italy took the Wooden Spoon for the 13th time even if they had managed a win at the Stadio Olympico.
Man of the match: Matteo Minozzi deserves a mention while Jake Polledri's work at the breakdown added value to the side. Meanwhile, Scotland's Stuart Hogg was the best for his side and his second-half try came at a valuable time. However, our nod goes to Italy flyhalf Tommaso Allan for his amazing overall performance and great kicking skills.
Tries: Allan 2, Minozzi,
Cons: Allan 3
Pens: Allan 2
Tries:Brown, Barclay, Maitland. Hogg
Cons: Laidlaw 3
Italy: 15 Matteo Minozzi, 14 Tommaso Benvenuti, 13 Giulio Bisegni, 12 Tommaso Castello, 11 Mattia Bellini, 10 Tommaso Allan, 9 Marcello Violi, 8 Sergio Parisse, 7 Jake Polledri, 6 Sebastian Negri, 5 Dean Budd, 4 Alessandro Zanni, 3 Simone Ferrari, 2 Leonardo Ghiraldini, 1 Andrea Lovotti.
Replacements: 16 Oliviero Fabiani, 17 Nicola Quaglio, 18 Tiziani Pasquali, 19 Abraham Steyn, 20 Giovanni Licata, 21 Guglielmo Palazzani, 22 Carlo Canna, 23 Jayden Hayward.
Scotland: 15 Stuart Hogg, 14 Tommy Seymour, 13 Huw Jones, 12 Nick Grigg, 11 Sean Maitland, 10 Finn Russell, 9 Greig Laidlaw, 8 Ryan Wilson, 7 Hamish Watson, 6 John Barclay (captain), 5 Jonny Gray, 4 Tim Swinson, 3 Willem Nel, 2 Fraser Brown, 1 Gordon Reid
Replacements: 16 Stuart McInally, 17 Jamie Bhatti, 18 Zander Fagerson, 19 Richie Gray, 20 David Denton, 21 Ali Price, 22 Pete Horne, 23 Blair Kinghorn.
Referee: Pascal Gaüzère (France)
Assistant referees: Jérôme Garcès (France), Andrew Brace (Ireland)
TMO: Marius Jonker (South Africa)