The quartet hold a three-shot lead over the chasing pack which is led by South African Louis Oosthuizen, who recovered from an opening 77, to join Cameron Smith (69), Shane Lowry (70) and J.B. Holmes (71) on one-under.
Overnight co-leader Spieth carded a one-over 71 to remain in a share of top spot, while Grace and Johnson fired rounds of 70.
Day, who has been plagued by vertigo over the past two days, produced five birdies on the back nine to climb up the leaderboard.
It was an incredible achievement for the Australian to complete his third round, let alone post a score under par, one of only a handful of players to do so on Saturday.
The 27-year-old, who began the day three back, looked frail as he prepared for his round, and early bogeys at the second and fourth suggested that an increasing struggle was on the cards. However, having taken the turn at level par, Day showed tremendous strength and courage as he battled through the back nine, with birdies at the 10th and 11th either side of a bogey taking him back under par.
He was not done yet either, with three birdies in the last four holes seeing Day rise up the leaderboard while many were swiftly heading in the opposite direction.
Day looked increasingly fatigued, but he battled on, and was received with great applause by the galleries, no more so than when he sunk his birdie putt on the last.
Spieth cut a frustrated figure late in the day, with the 21-year-old failing to find the touch on the greens which has seen him top the PGA Tour putting charts.
The Masters champion produced an up and down front nine that three birdies and four bogeys as he struggled to find his rhythm. A bogey at the 11th would be the last of the day, but Spieth will be disappointed that a birdie at the 15th is all that he has to show for on a back nine on which he engineered a number of birdie chances.
Johnson too will feel that he didn't quite make the most of his round. The big-hitting American used his long range off the tee to good effect, but couldn't always capitalise on the greens. The par-four 16th encapsulated this well, with Johnson finding the green off the tee only to three-putt for a par.
Having taken the turn in 33 after notching up four birdies and two bogeys, Johnson produced a bogey-birdie-double-bogey run from the 11th to the 13th to drop back into the pack after carving out a two-shot lead at one point.
Grace was solid in his round of 70 which included three birdies and three bogeys. The South African looked particularly impressive on the front nine, but was forced to scramble somewhat on the inward nine.
It was an incredible turnaround for Oosthuizen who was paired with an off-colour Rickie Fowler and a woeful Tiger Woods on the opening two days. Sitting at seven-over after the first round, making the cut would have been a good achievement, but back-to-back rounds of 66 have seen the South African rise into contention.
The 2010 Open champion made use of a lob wedge rather than a putter for much of the day, and the move paid dividends, with five birdies and a lone bogey seeing him end the day on one-under.
Overnight co-leader Patrick Reed endured a round to forget, with double-bogeys at the second, seventh and 10th holes coupled with two bogeys and a single birdie seeing him drop back to two-over, before birdies at the 13th and 15th saw him claw back to level-par only to bogey the 17th.
It was another frustrating day for McIlroy, who continued to struggle with his putting in a round that required 31 putts. The Ulsterman missed a handful of straightforward putts which prevented him from gaining momentum in a round of even-par 70 that saw two birdies on the front nine cancelled out by two bogeys on the back nine.