Koepka, who shared the overnight lead with Norway's Viktor Hovland and Spain's Jon Rahm, returned a spotless five-under 67 to set a 12-under target for the afternoon wave that headed out under menacing skies.
Resurgent Jason Day had looked poised to challenge Koepka until he stumbled to a double-bogey, bogey, par, bogey finish for a disappointing 72 that put the Australian in the clubhouse seven behind.
While Rahm, winner of three events already this season, was always expected to contend at the year's first major, Koepka was not.
"I just take it one shot at a time, one hole at a time, whatever I got in front of me," said Koepka.
As Koepka was putting the finishing touches on his round, five-time Masters champion Tiger Woods was teeing off.