Smith, ably assisted by David Warner then later Mitch Marsh, dug in desperately as England pushed for a consolation prize after they relinquished the urn at the WACA in the third Test.
Australia will take an unassailable 3-0 lead to Sydney, where the five-Test series finale starts on January 4.
England's hopes of a dead-rubber victory were hampered by rain on day four that washed out approximately one and a half sessions.
But the visitors denied Warner a record-breaking century then dismissed Shaun Marsh with the final ball in Saturday's morning session, tucking into lunch with genuine belief they could force a result.
England failed to conjure a breakthrough with the second new ball after the meal break, with Root then struggling for ideas on a docile deck that hardly deteriorated at all.
Smith reached three figures after 259 deliveries. Two more balls and it would have equalled the slowest century of his Test career, achieved at the Gabba in the series opener.
The world's best batsman has now batted for 1882 minutes in this series.
Mitch Marsh reined in his natural aggression, scoring 29 runs from 166 deliveries.
To put those numbers in perspective, Marsh required 130 balls to bring up his maiden Test ton during the previous game in Perth.
Warner also showed uncharacteristic caution, taking 48 overs to compile a 107-run stand with Smith.
That was until the vice-captain was on 86 and threw away his wicket - and a chance to become the first batsman to score a hundred in both innings of four different Tests.
Warner lashed at the first delivery he faced from part-timer Root, with James Vince gleefully accepting the skied edge to create some nerves in Australia's dressing room.
Jimmy Anderson delivered 59 overs in the match, while Stuart Broad sent down 52 overs.
