Smith marched off unbeaten after a 275-ball knock of 102, having nudged Australia to 263 for four and a lead of 99 runs before the captains agreed to end the match midway through the final session.
Number six batsman Mitchell Marsh was as stubborn as his captain, scoring 29 from 166 balls to protect his wicket through the last two sessions, though the unyielding pitch may have been England's greatest enemy as it declined to offer their bowlers any encouragement.
With Alastair Cook finding form with an unbeaten 244, the tourists can perhaps claim a moral victory after their batsmen built a first-innings lead of 164 runs.
But they will still head into the fifth and final test 3-0 down in the series and desperate for a face-saving win at the Sydney Cricket Ground.
Smith, having scored an unbeaten 141 in Brisbane and 239 in Perth, came to the crease before lunch on day four, his team vulnerable at 65 for two.
He had to dig them out of another rough patch when England grabbed two late wickets early on Saturday to leave Australia wobbling at 178 for four.
The skipper brought up his century by cutting part-time legspinner Dawid Malan for a single, but there was no fist-pump or roar of celebration as in the previous matches.
The 28-year-old simply raised his bat dutifully to the sparse fifth day crowd before battening down again.
Rain fell during lunch but cleared to give England ample time to find a breakthrough as Australia resumed on 178-4.
The tourists laboured fruitlessly on an unresponsive pitch, however, as Smith and Marsh proceeded with discipline, content to bat time rather than set a fourth-innings chase.
It had all looked so promising for Root's men when they grabbed Australia's fourth wicket on the cusp of lunch to leave their opponents just 14 runs ahead with six wickets remaining.
Paceman Stuart Broad had dismissed middle-order batsman Shaun Marsh for four with the last ball before the interval but Smith has proved a constant thorn in England's side throughout the series.
Opener David Warner, who proved similarly irksome to the tourists in the match, soaked up 227 balls for his 86 before midday.
He brought up his slowest test fifty with a late cut for two off all-rounder Chris Woakes and appeared cruising to another ton after his first innings 103.
However, he handed skipper Root a present on his 27th birthday, top-edging the part-time spinner with an attempted slog-sweep to be caught by James Vince.
Marsh survived for most of the last half-hour of the session before he was caught behind by wicketkeeper Jonny Bairstow, who took a sharp one-handed catch moving to his left.
With his seamers garnering little movement off a stubbornly flat pitch, Root resorted to a series of funky field placements and had his bowlers persist with a line wide of off-stump to try to remove the Australian captain.
Smith was impervious to it all, playing straight and bunting the ball back to the bowler repeatedly.
Warner had played in a higher gear and stroked a sumptuous cover drive off rookie seamer Tom Curran for his seventh four to charge into the seventies before his brain fade against Root left him trudging off ruefully.
First Published: December 30, 2017, 12:00 PM IST