Australia may have lost the Ashes, at 3-1 down in a five-match series, but captain-elect Smith's unbeaten 110 saw his side to 376 for seven at lunch on the second day.
They would have been in an even stronger position but for off-spinner Moeen Ali taking two wickets in three balls in what became the last over of the session to dismiss Peter Nevill (18) and Mitchell Johnson (nought).
This was the 26-year-old Smith's 11th hundred in 33 Tests and his second this series after he made a career-best 215 in Australia's 405-run win in the second Test at Lord's.
Smith was 78 not out overnight and Adam Voges unbeaten on 47, with their stand then worth 101. England took the new ball as soon as possible and gave it to Stuart Broad.
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But the paceman, whose return of eight for 15 had stunned Australia at Trent Bridge was, as he had been on Thursday after England captain Alastair Cook won the toss, again fractionally too short and wide in length and line.
Voges cut Broad for four to complete a 99-ball fifty, including nine boundaries, that followed his second innings 51 not out in Nottingham.
But Voges was plumb lbw for 76 when an off-cutter from Ben Stokes kept low to end a fourth-wicket partnership of 146.
That was the last ball of the over and next ball 332 for four should have become 332 for five when Smith, chased a wide delivery from Finn and edged to wicket-keeper Jos Buttler.
It was poor cricket all round, as was the failure by umpire Aleem Dar not to spot the huge no-ball until he called for replay assistance. Finn, though, did have his 100th Test wicket when Mitchell Marsh (three) opened the face and was caught by Ian Bell at second slip.
Smith's quick single off Ali saw him to a 197-ball hundred, including 12 fours and a six. Two balls later, Smith went down the pitch and drove Ali for six over long-off.
But Ali, who on Thursday had dismissed David Warner (85), had Nevill well caught down the legside by Buttler before bowling Johnson after the left-hander played down the wrong line to a turning ball.