The pair, both 20, had tied after 54 holes - 36 over the New Course on Saturday and 18 over the Old Course yesterday - on 13-under-par 213s.
Meek, who has just completed her first year at the University of Missouri, started the final round two shots behind overnight leader Carnet but found herself "flying" over the first nine holes of the Old Course.
She was six under par for the first seven with an eagle three at the fifth and birdies at the second, third, fourth and seventh in six-under 32 to the turn. Carnet, out in one-over 39, was swamped by the tide in Meek's favour and her two-shot initial lead became a five-shot deficit with nine holes to play.
The French player turned the tables on the inward route which she covered in 34 to Meek's 39 and capped her comeback by holing a 10ft putt on the 18th green to go round in 73 to Meek's 71 and force a sudden-death play-off over the first and 18th holes.
The first two holes were halved but disaster struck the French player at the third hole (the second replayed) when she put her second into the Swilken Burn after Meek hit the green with a solid iron shot to win the hole and the tournament with a par four. "It's my best yet. No doubt about that," she said.
"Playing on the US college circuit for a year has certainly made me a better player, more able to cope with the pressure of a play-off like this."
Meghan MacLaren (Wellingborough), a Curtis Cup team candidate until late in the race, finished third on 217. Alice Hewson's final round of nine-under-par 67 was the lowest of the day, only one shot outside the women's amateur course record.
Hewson (Berkhamsted), a Junior Vagliano Trophy player for GB & I, won the Lawson Trophy for the lowest total by an under-18 player.