Stefani took full advantage of a faltering third round from overnight leader Harris English to go four under for the day and lead on 12 under overall.
Harris started the morning with a two-shot lead but initially squandered it to Stefani before reclaiming it heading into the final holes.
He was unable to do better than a one-under score for the day, though, meaning Stefani's four back-nine birdies counted.
Harris can count himself lucky the difference is not greater too, with Stefani keen to forget a horror seven on the par-three 11th which, if played differently, could have put the tournament to bed.
As it is, though, it remains anyone's to win, with a clutch of well-placed names just off the pace waiting to pounce thanks to some low-scoring rounds.
Phil Mickelson and Padraig Harrington were among them, the multiple-major winners tucked in on seven and six under respectively and ready to make a move if the pace-setters stumble tomorrow.
Both trimmed five shots off the par today, while others such as Patrick Reid (six under and eight overall) and Nicholas Thompson (four and eight) are firmly in the mix. Scott Stallings (eight under) and Eric Meierdierks (seven) make up the lead group.
Ian Poulter enjoyed a successful day with a tidy two under and at five behind overall remains an outside chance, with the same applying to Dustin Johnson.