Burns' innings and his record 308-run partnership with Head (161) on the first day put Australia in an nearly unbeatable position, which could ensure a first Test series win since the 2017/18 Ashes.
Sri Lanka head coach Chandika Hathurusingha provided a positive update on opener Dimuth Karunaratne, after he was taken to hospital having been struck on the neck by a Pat Cummins bouncer in the second Test with Australia.
The second Test between Australia and Sri Lanka at Manuka Oval has been going the home team's way.
Paine opted against the follow-on in Australia´s last Test batting opportunity before the Ashes tour to England later this year.
The tourists reached 123-3 at stumps and trail Australia by 411 runs, with Kusal Perera unbeaten on 11 and Dhananjaya de Silva on one.
Perera ducked into a ball from Richardson in the first session of day three and received treatment on the field, with play stopping for several minutes.
The neck protector flew off and a physio rushed to his aid.More news: Here's how the Patriots Super Bowl roster evolved around Tom Brady
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Patterson, who smacked 14 fours and a six in his knock, and Paine then continued to plunder runs at will with an unbroken sixth-wicket stand of 130.
Ironically, the courageous Karunaratne replaced him after being declared fit again after being felled by a brutal bouncer on Saturday from Cummins.
A hush descended upon the sell-out crowd in Canberra as Karunaratne was tended to by medical staff from both teams and paramedics after being struck on the neck by a short ball from speedster Pat Cummins.
When they returned Dilruwan Perera and Ferndano lasted just eight more balls with Starc doing the damage, ending with 5-54 to silence his critics after a lean spell.
Highly-rated Australia batsman Will Pucovski, who will turn 21 on Saturday, has been released from the test squad playing Sri Lanka due to mental health concerns, the country's cricket board said on Friday.
Australia's batsmen had managed just one century in the past 12 months before arriving in Canberra, where Joe Burns (180), Travis Head (161) and Patterson ended the drought in emphatic fashion to stake their claims for Ashes selection.