The visitors had shown guts and determination to hold off continued French assaults in the opening 40 minutes but the pressure finally took its toll as Wesley Fofana and Maxime Medard crossed over for converted scores within four second-half minutes.
Scotland had led early from the boot of Greig Laidlaw and were tied at 9-9 as the Edinburgh scrum-half and French playmaker Frederic Michalak exchanged penalties before the home tries arrived.
Michalak had to leave the field with a serious shoulder injury after sparking France's second-half recovery, but the officials missed a blatant assault on Stuart Hogg earlier, which seems certain to be picked up by the citing panel.
A late Tim Visser touchdown gave the scoreline a more respectable look than was perhaps deserved but last year's wooden spoon winners can at least console themselves with a third-place finish in the championships - their best result since 2006.
Scotland had travelled to France hoping to register their best RBS 6 Nations finish since 2006 by finishing second - only to be denied before their match had even kicked-off as England's defeat to Wales kept them outside of the top two.
Meanwhile, given Italy's surprise win over Ireland, the hosts ended the championships rooted to the bottom of the table - a humiliation last suffered in 1999, the same year as Scotland's most recent win in the French capital.
Under-pressure Les Bleus boss Phillipe Saint-Andre went with power and bulk in his starting XV, restoring the wrecking ball that is Mathieu Bastareaud to his midfield while also bringing in flanker Antoine Claassen and lock Sebastien Vahaamahina for their first Test starts.
As for Scotland caretaker coach Scott Johnson, he made two changes to his team, bringing in debutant lock Grant Gilchrist for his first start while Perpignan forward Alasdair Strokosch also returned to starting duties.
With the wind and rain blowing down upon the Parisian suburb of Saint-Denis, conditions were not going to make the job easy for either side.
And with the downpours enough to sow the seeds of doubt, it was the visitors' Stuart Hogg who showed the first sign of nerves after two minutes of play when he knocked on a long punt into the Scots' 22, putting his side under immediate pressure.
Winger Sean Maitland saved any blushes with a brave tackle on Michalak and with that let off, Scotland pushed forward, gaining their just rewards after four determined drives into French territory when Laidlaw knocked over the opening penalty of the afternoon following an indiscretion by the home side at the breakdown.
Yet more handling errors from both sides highlighted the challenges presented by the weather but Laidlaw had adapted well to the circling gusts and doubled Scotland's lead with a second penalty.
Scotland's previous match, the 28-18 defeat to Wales last week, had seen the side struggle at the scrum and France knew that with their physically dominant squad, the set-piece offered them opportunity to exploit.
And it was with a concerted effort that they pushed themselves to within inches of the score line after 23 minutes, only for Jim Hamilton's last-ditch heroics to halt a powerful heave from the hosts.
Vincent Clerc was then brought down as the winger looked poised to dance his way over while yet more scrambling was required from the away side to prevent Bastareaud from battering in for a try.
It was exhausting stuff for Scotland but it was France who ran out of patience and let their under-pressure opponents off the hook with an infringement at their umpteenth try-line ruck.
Hogg had started the Six Nations in electric form but after two average displays against Ireland and Wales, the Glasgow man was looking more like his old self. Two rapid bursts from deep were just about held up by France and he was unlucky with a drop-goal attempt from distance just before the break.
But the argy-bargy between Hogg and Michalak three minutes after the restart did enough to unsettle Scotland and the French party to that dispute slotted over the home side's first points from a penalty following a collapsed Scots scrum.
When he was not tangling himself in needless squabbles, Michalak showed he could play a bit. His guile at fly-half was becoming ever more apparent as he teased the visitors one way and the other.
Again the Scots were forced into desperate defence and stepped over the line at the breakdown, handing the French number 10 the chance to slot a second kick over the posts to tie the scores.
The pressure was beginning to build on the Dark Blues and a third penalty was dispatched by the Toulon playmaker to put the French ahead for the first time.
In Laidlaw, though, Scotland have their own Mr Reliable and the Edinburgh scrum-half soon restored parity with his third penalty of the evening.
And yet French hunger for the opening try was not diminished. Neither, though, was the tenacity of Hogg as he stood brave to block the way of Clerc as he burst through the gain-line, holding up the galloping winger single handedly just yards from disaster.
But he could not repeat the trick as another overload on the right flank proved fateful.
Wesley Fofana broke through before brushing off Hogg's despairing lunge before running in to score under the posts with 14 minutes left. Michalak added the extra two points to rub salt into freshly-opened Scottish wounds.
Having defended for their lives for most of the match, the tank was now empty and France added a second touchdown with 10 minutes remaining.
Quick ball had been a rarity for the hosts for most of the game thanks to the sheer awkwardness of Scotland but there was little resistance as Maxime Medard collected from a ruck and skipped in between the sticks.
With Michalak ruled immobile, substitute Maxime Machenaud took over kicking responsibilities with a successful conversion to extend the team's lead to 14 points.
That lead was trimmed back to seven, however, as Scott darted forward before playing in Tim Visser for an easy try, with substitute Ruaridh Jackson converting.
There was little to worry about for France though and they counted down the five minutes that remained with little difficulty to finally collect a long-awaited Six Nations win.