Set 504 to win, England resumed on 5-251 and reached 6-332 at lunch on Tuesday's fifth and final day of play.
While it still seems unlikely, England now find themselves still in the hunt and needing 172 runs to claim one of the most remarkable wins in the game's history.
Matt Prior (26) dug in, adding 76 for the sixth wicket with Stokes.
The second new ball was always going to be a pivotal moment and Mitchell Johnson (2-70) thudded a short ball into Prior's ribs with the fourth over with the new nut.
Prior was out caught behind to a wide ball in Johnson's next over.
The courageous Stokes top-edged a bouncer from Johnson to the fine-leg boundary to notch his maiden hundred off 159 balls including 16 fours in his second Test.
Five Australia batsmen have scored a total of seven hundreds while the 22-year-old Stokes is England's only century-maker in the series.
Ryan Harris hit one of the large cracks on the pitch in the 73rd over as the ball jagged about a metre wide of Stokes' bat. Spinner Nathan Lyon has also found the cracks.
But England have resisted. Tim Bresnan is seven not out.
Stuart Broad is due to come in next for England, despite a bruised foot suffered while batting in the first innings.
The world record successful fourth innings run chase is the 7-418 scored by West Indies against Australia in Antigua in May 2003.
Australia put themselves in a strong position on Monday as Shane Watson smashed 103 in the home side's second innings of 6-369 declared.
"It's a special moment in everyone's career in the Australian team to be in the position that we are," Watson said on Monday night when victory appeared close at hand.
"We have to bowl really well to give ourselves the best chance to be able to achieve a few dreams, for a lot of us who've been involved in the last few series especially."
Watson is a veteran of Australia's past three losing Ashes series.
Australia lead the five-match series two-nil.
AUSTRALIA FIRST INNINGS 385
ENGLAND FIRST INNINGS 251
AUSTRALIA SECOND INNINGS 6(dec)-369
ENGLAND SECOND INNINGS
A COOK b Harris 0 1 1 0 0
M CARBERRY lbw Watson 31 (53)
J ROOT c Haddin b Johnson 19 (88)
K PIETERSEN c Harris b Lyon 45 (57)
I BELL c Haddin b Siddle 60 (93)
B STOKES not out 117 (183)
M PRIOR c Haddin b Johnson 26 (72)
T BRESNAN not out 7 (29)
Sundries (9b 13lb 5w) 27
Six wickets for 332
Fall: 0 (Cook), 62 (Carberry), 76 (Root), 121 (Pietersen), 220 (Bell), 296 (Prior).
Bowling: R Harris 19-2-73-1, M Johnson 22-6-70-2 (1w/5), N Lyon 18-4-61-1, P Siddle 26-11-67-1, S Watson 11-1-39-1.
Batting time: 422 mins. Overs: 96.
Umpires: Billy Bowden (NZL), Marais Erasmus (RSA).