Resuming on 3-235, Australia declared before lunch on Monday's fourth day of play on 6-369.
The world-record successful fourth-innings run chase is the West Indies' 7-418 against Australia in Antigua in May 2003.
Australia lead the five-match series two-nil and victory in Perth would allow Michael Clarke's men to regain the urn.
Watson smashed a 106-ball century before departing in bizarre circumstances for 103.
After resuming on his overnight score of 29 off 66 deliveries the talented right-hander went on to club 103 off 108 deliveries before top-edging a simple catch to cover which was dropped by England's Ian Bell with the score on 331.
Watson however had been ball-watching and didn't make his ground at the bowler's end as he failed to beat the throw from the bowler Tim Bresnan.
It was Watson's fourth century in 49 Tests. The No.3 batsman has been heavily criticised for his poor conversion rate which includes 21 half-centuries.
Watson has scored two hundreds in 2013, the first of which was in the dead-rubber fifth Ashes Test at The Oval in August.
The powerfully built 32-year-old helped himself to 22 runs off the 77th over of the innings on Monday as offspinner Graeme Swann was hit out of the attack. Watson's three sixes and one four in the over were all driven down the ground as he advanced to 82.
Bresnan had taken an overhead catch at long off when Watson was on 90 but the fieldsman tumbled over the rope and a six was declared in an action-packed morning session.
Watson leg-glanced a boundary later in the same over from Jimmy Anderson to reach his century and thrust his arms into the air in celebration.
His second 50 came off 28 balls.
Steve Smith made 15 before George Bailey smashed an unbeaten 39 off 30 deliveries.
Watson was Australia's second century-maker of the innings after opener David Warner's 112 against an attack depleted by the foot injury to pace spearhead Stuart Broad.
Broad has been cleared of a foot fracture and is expected to bat in England's second innings but did not bowl in Australia's second innings after receiving a blow while batting on Sunday.
South Africa scored 4-414 in 2008 in Perth in the second-highest successful fourth-innings run chase in Test history. However that WACA pitch was a docile surface in comparison to the cracked strip in use for the third Ashes Test.
AUSTRALIA FIRST INNINGS 385
ENGLAND FIRST INNINGS 251
AUSTRALIA SECOND INNINGS
C ROGERS c Carberry b Bresnan 54 (135)
D WARNER c Stokes b Swann 112 (140)
S WATSON run out (Bresnan) 103 (108)
M CLARKE b Stokes 23 (53)
S SMITH c (sub) b Stokes 15 (50)
G BAILEY not out 39 (30)
B HADDIN c Swann b Bresnan 5 (6)
M JOHNSON not out 0 (0)
Sundries (8b 5lb 5w) 18
Six wickets (dec) for 369
Fall: 157 (Rogers), 183 (Warner), 223 (Clarke), 301 (Smith), 331 (Watson), 340 (Haddin).
Bowling: J Anderson 19-5-105-0 (1w/5), T Bresnan 14-3-53-2, B Stokes 18-1-82-2, G Swann 27-8-92-1, J Root 9-1-24-0.
Batting time: 355 mins. Overs: 87.
Umpires: Billy Bowden (NZL), Marais Erasmus (RSA).