The Scots took the lead in the sixth minute through Leanne Ross and were holding on comfortably until their defender Jenny Beattie sent the ball past her own goalkeeper, Gemma Fay.
That gave France, ranked fifth in the world and far too good for Scotland in the Euro 2013 qualifiers, a reprieve and some reward for the possession they had.
The result is a significant lift for Scotland before they begin their bid to qualify for next year's World Cup in Canada with a double-header of matches against Poland and Bosnia-Herzegovina next month.
The Scots are in the same qualifying group as Sweden, whom the French beat 3-0 in their previous game.
"This shows we can compete with the best teams now," said Scotland's coach, Anna Signeul, after the evening game in Larnaca.
"France have been so good in recent games but we now have the strength and speed we have been missing before.
"The players showed real determination and their workrate was great. They really wanted to get a result. France had their best team out and did everything they could to get a goal and get back in the game.
"It's just a pity we had to concede the own goal which prevented us getting the win."
Scotland scoring first was not in the script, but good work by Kim Little and Hayley Lauder set up midfielder Ross for an easy goal.
France were at their most dangerous just after the goal and in the five minutes before half-time, but Fay had only two real saves to make in the entire match.
This was largely due to Signeul playing Rachel Corsie, who is normally a central defender, in front of her usual partner Beattie and Ifeoma Dieke.
The latter was surprisingly given her first international start since rupturing her anterior crucial ligament playing for Team GB at the London Olympics 20 months ago.
The three were outstanding in preventing a high-scoring French side from finding the net through their own efforts, the Scotland net only being breached when a Gaetane Thiney cutback led to Beattie's own goal.
Scotland's next game at the tournament in Cyprus is against the Netherlands tomorrow.