Bird cleaned up the final wicket of the day Suranga Lakmal for five at stumps to finish the determined Sri Lankan innings.
Fellow Aussie quick Mitchell Starc ripped through the tourists collecting three wickets, while veteran Peter Siddle snared two scalps.
The Australians battered the tourists relentlessly on the green Sydney pitch, after winning the toss and electing to field.
At stumps on day one, Sri Lanka was all out for 294, with Australia to start the chase on day two in what is veteran champion Mike Hussey's final Test.
Sri Lankan batsman Lahiru Thirimanne top-scored for the tourists with a hard-fought 91, while departing skipper Mahela Jayawardene made a valuable 72.
But, the tourists slowly crumbled under the weight of their injury crisis and a fired-up Australian pace attack, with wickets falling constantly despite their determined fight.
Australia's bowlers didn't knock over the top order as quickly as they would have liked, but rallied in the final session to take 5-72 and reclaim the advantage.
Controversially rotated out for the Boxing Day Test, Starc responded with 3-71 including the big breakthrough of Jayawardene caught at first slip in the middle session.
Unwanted by NSW as a young fast bowler, Bird has gone on to star in Sheffield Shield cricket with Tasmania and is doing the same two matches into his international career.
Bird (4-41) removed openers Dimuth Karunaratne and Tillakaratne Dilshan before knocking over tailenders Rangana Herath and Suranga Lakmal for his first four-wicket Test haul.
Siddle (2-46) toiled away well, but Mitchell Johnson (0-58) and Nathan Lyon (1-69) battled.
Thirimanne combined with Jayawardene and Mathews for important half-century partnerships before the Test rookie threw his wicket away to be caught by a diving David Warner off Lyon late in the day.
It looked as though Australia would have to face a few overs at the death, but tailenders Lakmal (5) and Nuwan Pradeep (17no) put on a 21-run stand that lasted long enough to ensure there was no time for Australia to bat on the opening day.
- With AAP