Duncan Weir kicked a last-ditch drop-goal to salvage a precious victory for Scotland over Italy in Rome.
Italy led 13-3 at the break, Tommaso Allan scoring a try for the hosts.
The visitors had more purpose after the restart and Alex Dunbar crossed twice in 13 minutes to give Scotland a five-point lead with 13 minutes remaining.
Italy hit back immediately courtesy of a converted try by lock Josh Furno, only for fly-half Weir to clinch the win for Scotland with 15 seconds left.
It sealed the most dramatic of victories, ending Scotland's run of eight consecutive away defeats in the Six Nations and bringing the Scots their first points of the campaign. Italy are bottom of the table with three defeats.
And apart from a 20-minute spell at the end of the first half, his charges did just that.
The overarching issues remain and the enquiry into the wisdom of Johnson's long-term involvement as director when Vern Cotter takes over will continue, but for now this is a reversal of recent fortunes to be savoured.
Scotland's line-out curse appeared to have transferred itself to Italy with the sun beating down in the opening 10 minutes at the Stadio Olimpico.
Lock Jim Hamilton read the movement well and was twice on hand to steal the white-stripped hosts' set-pieces, while the recalled Richie Gray was imperious at collecting those from his own side.
It was one of those successes that led to a Tommy Seymour charge down the right but the Glasgow Warriors man over-hit his chipped kick.
A charged-down kick by the Italians lifted the home crowd, however, and Scotland were about to enter a frustrating spell of collapsing scrums and a spiralling count of conceded penalties - by the 37th minute the Scots had given away 10 to Italy's two.
Full-back Luke McLean initiated Italy's first meaningful attack, which saw Allan crossing the line only to realise referee Steve Walsh was bringing play back for a penalty.
But for the visitors, the irony of being scored against by their own former Under-20 stand-off was only delayed, as Allan picked himself up to put the first points on the board.
At times, Johnson's men worked through the phases and retained possession competently, albeit without a killer pass - highlighted by Alessandro Zanni's block on a Weir offload to end a passage of play that looked like it might bring Scotland's first try of this Six Nations campaign.
Laidlaw drew Scotland level with a well-hit penalty but it only preceded a disappointing end to the first half.
The Italians clearly had the physical edge in the scrum and that upper hand was soon to manifest itself in more scoring.
Allan's penalty made it two successful kicks from three before Sergio Parisse - who along with Martin Castrogiovanni became Italy's joint most-capped player on 104 - displayed a powerful piece of play to bring the hosts a try.
His offload to Furno was recycled to Allan, who rubbed further salt in the wound by crossing over near the posts. A straightforward conversion gave Italy a comfortable 13-3 half-time lead.
As they had the first, Scotland started the second half brightly and Laidlaw's penalty preceded a period of concerted pressure of which Gray, trying to show he is the man to build a team around, was frequently the hub - and it eventually paid off.
Scott Lawson and Stuart Hogg's offloads were perfect to present Dunbar with the opportunity to burst between two Italians and touch down in the corner.
Laidlaw sent a difficult conversion wide but the Scots long search for an away win in this competition looked increasingly likely to come to an end in the Eternal City.
Dunbar and Matt Scott excelled as Scotland continued to press, taking advantage of the Italians' fading intensity.
Allan's first-half scores for Italy may have been tough for the Scottish hierarchy to watch, but it was the youngster the visitors targeted as the weak link in defence, a tactic that worked when Dunbar crossed the line with at least one Italian hanging onto his ankle.
Replacement Chris Cusiter was instrumental in the build-up to the score, which left the turnaround seeming complete with 10 minutes remaining.
However, the influential Parisse once again charged forward and helped Italy get in behind the visitors, exploiting Scotland by working the ball out to Furno to bring his side level out wide.
Luciano Orquera scored what looked like the winning conversion but Scotland's persistence brought a scrum, and when the ball was fed to Weir, he secured victory with seconds remaining.
Italy: McLean, Esposito, Campagnaro, Garcia, Sarto, Allan, Gori, De Marchi, Ghiraldini, Castrogiovanni, Geldenhuys, Furno, Zanni, Barbieri, Parisse.
Replacements: Giazzon, Aguero, Cittadini, Bortolami, Derbyshire, Botes, Orquera, Iannone.
Scotland: Hogg, Seymour, Dunbar, Scott, Lamont, Weir, Laidlaw, Grant, Lawson, M. Low, R. Gray, Hamilton, Wilson, Fusaro, Beattie.
Replacements: Ford, Dickinson, Cross, Swinson, Denton, Cusiter, Taylor, Evans.
Referee: Steve Walsh
Touch judges: Jerome Garces and Luke Pearce
TV: Geoff Warren