Rahm birdied the par-4 third and par-5 eighth holes but made bogey at nine to reach the turn at 10-under for the tournament while Koepka made bogeys at the par-3 fourth and sixth and the ninth to stumble two back after leading by two through 54 holes.
When Rahm missed a six-foot par putt at the ninth, the stage was set for a dramatic fight down the stretch for the green jacket in windy conditions under the Georgia pines as leaders stumbled while challengers caught fire.
The Masters champion will pocket a record top prize of $3.24 million from a record $18 million purse.
Spain's Rahm, the 2021 US Open champion with three PGA Tour titles this year, chased his first Masters win on what would have been the 66th birthday of his idol, the late Seve Ballesteros, who won the Masters 40 years ago.
"I'd like to think he's up there watching and pulling for me," Rahm said. "If there's somebody who would have enough charisma to give me any help, it would be him."
World number three Rahm could become the fourth Spaniard to grab the green jacket after Ballesteros, Jose Maria Olazabal and Sergio Garcia.
Rahm also would overtake defending Masters champion Scottie Scheffler as world number one with a victory if the American finishes outside the top four. Scheffler was 4-under overall with four holes to play in a share of 14th.
Koepka, a four-time major winner, led on 11-under 205 after 54 holes with Rahm two back after both concluded one-over 73 third rounds Sunday morning. Heavy rains Saturday had halted play due to puddling on greens.
In the final round, Rahm sank a 10-foot birdie putt at the third to reach 10-under and Koepka found a bunker at the par-3 fourth then missed a nine-foot par putt, leaving the two level for the lead.
Koepka missed the green at the par-3 sixth and a poor chip 22 feet beyond the hole led to a bogey that put Rahm into the lead and the Spaniard dropped his approach inches from the hole at the par-5 eighth to set up a birdie that lifted him to 11-under.
On the day, Koepka had gone 19 holes without a birdie after sitting on a four-stroke edge overnight.
Koepka, coming off a LIV Golf victory last week in Orlando, has won all three prior times when leading a major after 54 holes.
A marathon 30-hole Sunday was on tap for the final duo. Neither Rahm nor Koepka, who played 12 morning holes, had a back-nine birdie on the first try, giving hope to a world-class field in pursuit.
The greatest last-round Masters comeback to win was eight strokes by Jack Burke in 1956, but several were in contention to break that mark.
Three-time major winner Spieth, who began the round 10 off the pace, reeled off three straight birdies starting at eight, two more at 14 and 15 and another at 17 to share second with Koepka on eight-under.
Six-time major winner and three-time Masters champion Phil Mickelson, trying to break his own major age win mark at 52, birdied the par-3 12th, par-5 13th and 15th and 17 to reach 7-under and share fourth with US countryman Russell Henley and Cameron Young. Mickleson would become the clubhouse leader at eight-under
On six-under were Americans Patrick Cantlay and Patrick Reed.
Koepka, Reed and Mickelson gave Saudi-backed upstart series LIV Golf three top contenders.
They were among top PGA Tour stars lured last year with record $25 million purses from 54-hole events, sparking the PGA to ban LIV players from its tournaments even as majors allow LIV golfers to compete.
