Just as she had done on Saturday, the 21-year-old American carded eight birdies at Magnolia Grove - spoiled only by a bogey at the par-four fifth.
Her 21-under-par total saw her finish a shot ahead of compatriot Jessica Korda (68) and Thailand's Pornanong Phatlum (63).
South Korean duo Jiyai Shin (63) and Chella Choi (70) were a further stroke adrift, tied for fourth alongside Thailand's Aryia Jutanugarn (65), Sweden's Anna Nordqvist (69) and Australia's Karrie Webb (68). Former world number one Stacy Lewis of the United States was three shots off the pace after a final-round 67.
Johnson, who holed a 15-foot birdie putt on 17 to move to the brink of victory, told www.lpga.com: "I'm a little shocked. I didn't even realise I shot 65.
"The whole back nine was kind of like... I don't know, I don't know what was happening.
"The birdie putt on 17, that's when I started to think a little bit more about winning the tournament, on 17. A little bit on the front nine, but you can't afford to think about that."