At the close of a stop-start day through showers, the Windies were 207 for six with Bravo offering spirited resistance on 94 with Kemar Roach in support on 31.
Bravo and Roach kept the Windies innings ticking over with an unbeaten seventh-wicket stand of 91.
Lyon claimed two wickets in one over to trigger a middle-order collapse to have the West Indies in familiar trouble in their chase after a mammoth declaration of 583 for four on the second day.
The probing off-spinner snared the wickets of Marlon Samuels and Jermaine Blackwood in his seventh over along with opener Rajendra Chandrika for 175 Test wickets to leave the hapless Caribbean tourists struggling to avoid a heavy defeat with three days left.
The Windies sticky predicament was accentuated by the dismissal of skipper Jason Holder, the last of the recognised batsmen at the crease with Bravo.
Holder refused to seek a referral after being struck high on the pad by Peter Siddle and walked off for 15, leaving his side in deep strife at 116 for six.
Replays showed that the ball was missing the stumps and the towering Holder may have stayed on as the players left the field for a brief rain break.
The West Indies efforts were in sharp contrast to Australia's run spree, propelled by a record 449-run fourth-wicket stand by Adam Voges and Shaun Marsh.
The West Australian pair eclipsed the fourth-wicket Test record of 437 set by Sri Lankans Thilan Samaraweera and Mahela Jayawardene against Pakistan in 2009.
It was also the biggest Test stand by Australians at home, bettering the 405-run fifth-wicket partnership by Sid Barnes and Don Bradman against England in Sydney in 1946.
But the duo fell two runs short of Australia's all-time highest partnership for any wicket of 451 held by Bill Ponsford and Bradman against England at The Oval in 1934.
Voges also posted Australia's highest score against the West Indies with an unbeaten 269, eclipsing Doug Walters' 242 in 1969.
Marsh holed out just before lunch for 182, caught on the ropes by Bravo off spinner Jomel Warrican, ending a day's occupation of the crease as he compiled his highest Test score.
Australia hammered 145 runs in the session to go to lunch at 583 for four enabling Steve Smith to declare the innings during the break.
The Windies made a cautious start after lunch, but soon began to unravel.
Kraigg Brathwaite lasted 26 balls before Josh Hazlewood trapped him leg before wicket for two in the ninth over.
Chandrika followed in the 20th over when he was deceived by a drifting Lyon delivery and got an outside edge to Smith at slip for 25.
Samuels lasted only 14 balls before he was splendidly caught and bowled by Lyon for nine.
Blackwood didn't see out Lyon's over and was gone five balls later, caught at bat-pad by Joe Burns, giving the spinner three for 18.
The Windies attack went down a bowler when quick Shannon Gabriel left the field with what a team spokesman said was a stress reaction in his left ankle, putting him in doubt for the rest of the series. – AFP
Original source: Lyon has Windies in a spin