With birdies proving easy to come by in Sunday’s third round, Dunne made six of them in a bogey-free round of 66 to reach 12 under with 18 holes remaining in the weather-delayed tournament.
Dunne was one of five amateurs to make the cut and he ended his third round having only dropped two strokes all week, one apiece in his matching opening two rounds of 69.
He started Sunday with a birdie at the first and picked up three more strokes on the front nine, with a fifth birdie of the day coming at the 10th as he closed on then-leader Danny Willett.
Dunne found life a little harder in the closing stages, but did manage to birdie the 15th and completed a couple of fine escapes to ensure a place in the final group on Monday.
No amateur has led the Open after three rounds since America’s Bobby Jones in 1927 – he is also the last amateur to win the title, a victory that came three years later.
Playing partner Oosthuizen was in similarly strong form, managing to better Dunne by making seven birdies, but he dropped shots at the eighth and 12th holes in a round of 67.
The 2010 St Andrews champion bounced back from his disappointment at the 12th with back-to-back birdies at 14 and 15, and the South African also produced a sublime approach to the last for a tap-in birdie.
Day, who looks completely recovered from the vertigo attacks he suffered in the US Open at Chambers Bay last month, joined Dunne in completing an error-free round with five birdies in his 67.
The Australian had a chance to take the outright lead on the 18th green, but his curling birdie putt slipped past the edge of the cup to leave him tied with Dunne and Oosthuizen, and one shot ahead of Grand Slam chasing Jordan Spieth, who made the most of ‘Moving Day’.
Spieth, already winner of the Masters and US Open titles this season, started the day a handful of shots off leader Dustin Johnson but quickly started to make inroads with a birdie at the first.
Further birdies followed at the fifth and seventh, before a dropped shot at nine saw him briefly show his frustration by hitting his own bag.
That outburst paid dividends immediately as he birdied the next three holes, and a further shot was picked up at the 15th to see Spieth round in 66.
But it was a bad day for Saturday’s leader Johnson, who made his task on Monday much more difficult with a three over round of 75, only one shot better than the worst round of the day.
Johnson struggled on the greens all day and made 13 pars in his first 14 holes, added to a dropped shot on the seventh.
He finally made a birdie, putting from the fairway on the 15th, but that preceded a nightmare finish with bogeys at the final three holes, including one at the last where he drove within 50 yards of the green only to leave his sloppy approach in the Valley of Sin and three putt from there.
Willett also struggled on the back nine, having made it to 12 under when he made his third birdie of the day on the 10th hole.
But the Sheffield professional dropped shots at the 13th, 14th and 17th to find himself three shots off the pace on nine under.
There was little to cheer for patriotic Scottish fans in the crowd as both Marc Warren and Paul Lawrie failed to stay in challenging positions.
Warren (-7) had two birdies in his first 11 holes but dropped shots at 12 and 15 to stumble round in level par 72, while Lawrie’s 74 included a double bogey at the Road Hole and left him six shots off the pace with 18 to play.
But there were plenty of big movers further down the field, with double champion Padraig Harrington climbing within two shots of the lead thanks to a seven under 65.
Harrington, back-to-back Open winner in 2007 and 2008, stormed through the field and gained 20 places thanks to his seven birdie round, one that gave him a decent chance at winning a third Claret Jug.
His round was not best of the day, though, with that honour belonging to Australia’s Marc Leishman, who gained 44 spots with an eight under 64 – all eight of his birdies came in 12 holes from the fourth hole.
Leishman sits in a tie for sixth place, three shots off the lead on nine under, a group which also includes a second amateur player – America’s Jordan Niebrugge, who had six birdies in a round of 67 – as well as England’s Justin Rose (68).
Another Englishman Eddie Pepperell appeared to be on for a record day among the earliest starters as he kicked off his round with five birdies in the first six holes.
Pepperell was eight under for the day when he reached the Road Hole but he lost concentration for a split second and carved his drive way right into the Old Course Hotel – a double bogey ensued but he is in touch for a final day charge on eight under thanks to his 66.