But despite Bird's best efforts New Zealand resurrected their drive to save the contest as BJ Watling and Matt Henry stretched the second innings to 311 for 7 at tea, a lead of 176.
Watling, who has five hundreds to his name, was on 46 with Henry on a Test-best 45 in a 101-run partnership.
Williamson was out for 97, denied his century after he and Corey Anderson had batted stoically through the morning session before Bird's breakthrough reduced New Zealand from 207 for 4 to 210 for 7.
With the new cherry in hand Bird bowled Williamson with a full-length delivery that zipped back and two balls later found the edge of Southee's bat to give Steve Smith the catch at second slip.
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Williamson, who spent 340 minutes in the middle, added 102 with Anderson for the fifth wicket to lay the groundwork for New Zealand's outside chance of saving the Test.
When Southee fell second ball, Watling and Henry teamed up to keep the innings going.
Where Williamson and Anderson had been the epitome of caution, Watling and Henry took the aggressive route in an attempt to force the Australian bowlers, who were working on reverse swing, off their line.
Williamson was the wicket Australia wanted most and emotions boiled over in the last over before lunch when a Josh Hazlewood lbw appeal was rejected.
After the Australian review failed with evidence of bat on ball, the Australian players converged on the umpires and the stumps microphone picked up an obscenity from Hazlewood.
Australia's stated aim before the Test was to win or at least the draw the Test which would put them on top of the world rankings.