Americans Blayne Barber (64) and Daniel Berger (68) were at 15 under, with Canada's David Hearn (67), New Zealander Danny Lee (68) and Zimbabwean Brendon de Jonge (68) a further stroke back in a three-way tie for fifth.
Former U.S. Open champion Rose, a runner-up at the Masters this month, piled up seven birdies in a flawless display to hold the clubhouse lead at TPC Louisiana before being joined by Day.
Australian world number six Day, the highest-ranked player in the field, drained monster putts for birdie at the sixth and 14th holes but missed several other birdie attempts from inside 10 feet.
With organisers hoping to conclude the tournament on a marathon Sunday, there was very little time for the players to rest before they set off in the final round.
"Back out in 30mins for the fourth round!" tweeted Rose, who is bidding for his seventh victory on the PGA Tour.
Slugger White, the PGA Tour's vice president of rules and competitions, said the goal was to complete the 72 holes as scheduled but added, "we would be here Monday if we have (lengthy) delays".
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Gene Cherry)
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