Lincicome, who started four shots behind Lorena Ochoa and Laura Davies, survived a blustery day and a six-hour round to shoot an even-par 72 - enough to give the 21-year-old her second career LPGA Tour victory.
She finished at 10 under, one shot better than Ochoa (77) - who was 6 over on her final six holes, a stunning collapse that cost her an opportunity to pass Annika Sorenstam for the top spot in world rankings.
Only a win would have pushed Ochoa to No. 1, but the Mexican star missed a 10-foot bogey try on the final hole that would have forced a playoff.
'It's hard to describe,' Ochoa said. 'Not meant to be. I tried really hard.'
The 21-year-old Lincicome earned $390,000 for the win and, on a day where only one player - Meena Lee, who shot 71 - broke par, even she couldn't believe what had transpired.
'When the wind picked up, even par was looking pretty good,' said Lincicome, the Women's World Match Play winner last year.
Davies (79) had the outright lead with two holes to play, but three-putted her way to double bogey at the 17th and then bounced around bunkers for a disastrous triple bogey at the last. She finished alone in third at 7 under, one shot better than Juli Inkster (72) and Nicole Castrale (74).
Conditions were simply brutal. Morning storms delayed play for 2 1/2 hours, and once the sky cleared, the wind followed. A 41 mph gust was measured in the area Sunday afternoon, and the wind steadily blew in the 20-25 mph range, causing an already tough course to nearly become unbearable.
'The sand would come out of the bunkers and just hit us,' Ochoa said, tapping her face. 'We were talking about how hard the wind needs to blow before they stop play.'
Players backed off most every putt, often two or three times. Ochoa was rattled at the 16th when she stepped away from her par putt twice, the second time because her cap was about to blow off her head. And play was dreadfully slow; play resumed at 12:15 p.m., and it was a few minutes past 6 p.m. when Ochoa missed her putt on the 18th, sealing Lincicome's victory.
'Tough conditions prove who's the best player,' Ochoa said. 'And today it was Brittany, for sure.'
The day's average score was 76.5, 4 1/2 shots higher than it was Saturday. Natalie Gulbis, who started the day tied for third and four shots off the lead, shot an 80 and finished 2 under, and 10 other players shot in the 80s.
The back nine was simply wild, with all three players in the final group getting some time atop the leaderboard.
Ochoa made the turn at 15 under, one shot ahead of Davies and four up on Lincicome - who actually was six shots back with 10 holes to play. But Ochoa made a double-bogey at 13 and bogeys at the 15th and 16th, giving back four shots in a four-hole span, a collapse that gave Davies a one-stroke lead with two holes to play.
And then it was her turn to waste an opportunity - with an epic collapse.
Davies' third shot at the par-5 17th went over the green. She chipped to about 6 feet, barely inside of where Lincicome's third shot from a fairway bunker 95 yards away landed.
While Ochoa made par, after getting her second roughly 30 yards from the flag, Lincicome's par try dropped perfectly in the center of the cup. Davies studied Lincicome's putt, knowing hers was on exactly the same line.
But she missed, and exacerbated her woes by having the bogey try from 3 feet lip out - making her settle for a double-bogey 7 and giving the lead back to Ochoa and Lincicome at 11 under.
At the 18th, things got even worse for Davies. Her tee shot landed in a fairway bunker, and her second shot clipped the lip of that trap and landed in another one a short distance ahead. Davies' third shot didn't reach the green either, she eventually made triple and saw her hopes of ending a nearly six-year drought between wins on the LPGA Tour come to a sudden end.