Rose had five birdies in his last six holes while Spieth made a complete hash of the last two after appearing to be in cruise control, with a lead as big as seven shots at one point late in his third round.
Spieth went into the day having made only one bogey in 36 holes, but his front nine was a struggle as he followed a birdie at the relatively easy par-five second with a dropped shot at the fourth.
A superb two at the sixth took him back to 15 under but he gave that shot straight back at the seventh before closing the nine with a birdie at the ninth, courtesy of a sensational approach shot to the tough, sloping green.
He moved to 16 under with a birdie at the short 12th, thanks to another superb short iron, but after picking up a further shot at the 13th he gave it straight back at 14 with a sloppy three putt.
But, just as Phil Mickelson threatened to eat into his lead, Spieth conjured up birdies at the 15th and 16th, another superb iron shot at the latter giving him the chance to move to 18 under.
However signs of nerves first arrived on 17, when his drive found the trees, and a duffed chip and three putt led to a double bogey six.
It looked as if the 18th would prove just as troublesome for the 21-year-old as he scattered the crowd with his approach to the green, but a sensational chip and eight foot putt meant a less than regulation par and a round of 70 – his 54-hole total of 200 is another scoring record, beating Raymond Floyd (1976) and Tiger Woods (1997) by one.
Three-time Augusta champion Mickelson received some of the biggest cheers of the day, especially on the front-nine when he had four birdies to make it to 10 under.
However he was only able to improve that position by one over the back-nine, with a bogey at the 17th meaning Rose will go into the final round alongside Spieth as his closest challenger – Mickelson signed for a round of 67.
Second position looked to be a long, long way off at the fifth when Rose fell to five-under with his second bogey of the day.
But the rest of his round was blemish-free, with birdies at the seventh and ninth getting him back to level par and setting up a fantastic back-nine run.
Rose had to wait until the 13th for his scoring streak to begin, kicking off a run of four successive birdies with a short putt at the par-five, and he reached 12 under with another birdie at the 18th, closing out a round of 67.
Charley Hoffman battled hard for a 71, with three birdies and two bogeys, to stay in contention in fourth place, six shots behind Spieth.
The leading quartet have a four-shot gap on the rest of the field, headed by a group of five players on six under that includes Rory McIlroy and Tiger Woods, who will play together on Sunday,
McIlroy and Woods both carded 68 on Saturday, although the current world No 1 was disappointed with his finish with bogeys at the 16th and 18th knocking him two shots further back from Spieth.