Smith went to his 25th Test century as he became only the fifth Australian – and the first since Matthew Hayden in Brisbane in 2002 – to register twin tons in the same Ashes match, following up 144 in the first innings with 142 in the second.
After Smith was eventually prised out Wade took centre stage, registering a Test-best 110 before Australia belatedly declared on 487 for seven, leaving them 30 minutes to bowl at the hosts on Sunday evening.
England's highest successful fourth-innings chase is the 332 they made against Australia in Melbourne in 1928, while they have never surpassed 315 batting last to win a Test on home soil, achieved against the same opponents at Headingley in 2001.
This was a day where the continued absence of James Anderson felt decisive. England's record wicket-taker will bat if and when required but the calf complaint he suffered on the opening morning precluded his involvement in Australia's second innings.
Despite prodigious turn on an increasingly worn surface, the expensive Moeen Ali was woefully out of sorts, and sent down two moon balls, while part-timers Joe Denly and England captain Joe Root were largely ineffective.
Australia started the day only 34 ahead with seven wickets left, and it was clear from that stage that the primary goal was to remove Smith as early as possible.
Moeen's first delivery to Australia's talisman kept devilishly low but the fifth was a looping beamer, signalled a no-ball which hardly engendered confidence at the start of the spell that yielded no chances.
Stuart Broad had two lbw shouts against Travis Head but neither was particularly convincing while Smith's uppish drive evaded the fielder at cover before racing for four, one of precious few misjudgements from the former Australia captain.
Both batsmen had passed fifty and Australia's lead was into three figures before Ben Stokes made the breakthrough, a short and wide off-cutter tempting Head into a cut that caught the edge and was pouched by Jonny Bairstow.
There was some speculation that a fraction of Stokes' back foot had landed on the return crease on the initial point of contact, which would have led to a no-ball being called. However, the umpires reviewed the footage and the decision stood.
Smith went to lunch unbeaten on 98 after declining to go for a big hit in the final over of the session but he was soon celebrating another hundred with a sumptuous cover drive for four off Broad.
England went for a fanciful review when he was on 125 and, though there were a couple of moments of fortune for Wade, Australia's pair settled for gentle accumulation to take the lead beyond 200.
England waited until the start of the 85th over to take the new ball and though Wade was able to overturn an lbw decision off Broad, Smith's famed patience finally snapped off Woakes – who had not bowled at all on Sunday morning.
The first legitimate delivery of the Warwickshire seamer's second spell was comfortably wide of off-stump but Smith, perhaps deceived by some late movement, loosely drove and was caught behind.
Wade, though, took advantage of a flagging attack after tea, moving to his first Test century in six years before he and Tim Paine departed in quick succession, the former pulling Stokes to deep square-leg and the latter bowled through the gate by Moeen.
The turn Moeen got for that wicket is a foreboding sight for England but their premier spinner was unable to take full advantage, bowling another beamer, as James Pattinson (47no) and Pat Cummins (26no) punished England further.