England replied with 374, their first-innings lead of 90 built on a maiden Test century from left-handed opener Rory Burns, who made 133, and fifties from captain Joe Root and vice-captain Ben Stokes.
Former Australia captain Smith's second-innings century and a maiden Ashes hundred from Matthew Wade allowed visiting captain Tim Paine to declare on 487-7.
Australia capitalised on the absence through injury of James Anderson, England's all-time leading Test wicket-taker, who only sent down four first-innings overs before breaking down with a calf problem that meant he did not bowl again in the match.
The match was the first of the International Cricket Council's new World Test Championship, with Australia's win giving them 24 points.
The five-match series continues with the second Test at Lord's starting on August 14.
Australia win first Ashes Test against England by 251 runs