Europe, who trailed 4-0 after Friday's opening session, needed to split the last-day singles at Finca Cortesin in Spain to gain a tie and retain the team trophy.
American Lexi Thompson beat Europe's Emily Pedersen in the final match to end a thrilling competition 14-14.
It was the first time that the matches have ended in a draw and the first time Europe has held the Cup three times in row.
Ciganda, all square with three to play, holed from four feet at the 16th and from two feet at the short 17th.
"I'm so happy," said Ciganda, who was congratulated by Spain's King Felipe VI, and in turn thanked European captain Suzann Pettersen.
"I love Suzann, I love Spain and I love the Solheim Cup," said Ciganda, who won all four of her matches.
Pettersen, who famously holed the winning putt at Gleneagles in Scotland four years ago, shed tears of joy.
"It was meant to be for Carlota in Spain," she said. "I said to her going down 16 is this what you would have wanted, and she said she was up for it."
Another home star was Caroline Hedwall who turned a three-hole deficit into a two-hole win over Ally Ewing.
With the pair all square with two to play, Hedwall sank a 30-foot birdie putt at the 17th.
It was an amazing turnaround for a home side that suffered a 4-0 drubbing on the opening morning foursomes.
"There is no such thing as giving up," said Pettersen. "We got off to a rough start, but turned it round and there will be a big party tonight."
The final day started in a tie.
Leona Maguire got the first point on the board with a decisive 4 and 3 victory over Rose Zhang. A star of Europe's win in Ohio as a rookie two years ago, the Irish woman helped clinch the point by holing a 35-foot eagle at the par four 14th.
"It feels just incredible," said the 28-year-old after nudging Europe into the lead for the first time. "Suzann gave me a job to do and the fans were just fantastic."
Momentum
The first point for the USA came in the top match with Megan Khang beating Linn Grant by one hole in a high-quality contest. The American was six under par in the tricky, windy conditions.
"I'm super proud," said Khang. "I knew that Linn would bring it and be a tough opponent."
Danielle Kang put America 10-9 ahead with a 4 and 2 win over Charley Hull, but Anna Nordqvist, Europe's most experienced player, tied the score again with a 2 and 1 victory over Jennifer Kupcho.
As the tension mounted the tide seemed to turn in America's favour.
Europe's Georgia Hall, who had been two up early in the back nine, three-putted the 17th from 12 feet to allow Andrea Lee to level the match.
At the same time, Europe's Gemma Dryburgh, who had been three up, missed a short par putt at the 16th ad also fell back to a tie with Cheyenne Knight.
Lilia Vu, who won 4 and 3 over Madelene Sagstrom, and Angel Yin, 2 and 1 over Celine Boutier, added two points for the Americans.
Hedwall and Ciganda switched the momentum and Thompson's 2 and 1 win over Pedersen in the final match was only good enough to make it 14-14.
Europe had kept the Cup.
