After Australia had been bowled out for 259 in their second innings to set West Indies the sixth highest run chase in Test history, the home side failed to get through the seven overs to lunch unscathed.
Barath clipped a Ben Hilfenhaus delivery off his legs into the air and Ed Cowan dived to his right to take a fine catch in the third over.
Only two runs were scored before lunch with Kraigg Brathwaite ending his run of three consecutive scoreless innings.
Australia, who have already wrapped up the series, started the day on 200 for six and looking determined to keep the scoreboard ticking despite the difficulty of scoring on the slow track.
Mike Hussey had taken his score to 32 when off-spinner Shane Shillingford found the edge and skipper Darren Sammy took a neat catch around ankle height at first slip.
Mitchell Starc decided that the aerial route was the best option against Shillingford.
The ball seemed to be either beating the edge or flying in the air off the bat towards the boundary as the pair duelled.
When Narsingh Deonarine came on, Ryan Harris tried to slog his second ball but it took the top edge and went high in the air where Carlton Baugh took it easily in his gloves with the batsman on nine.
Hilfenhaus, on six, then received a brute of a delivery from Shillingford that took a big bounce off the wicket and hit the splice of the bat.
Brathwaite, at silly point, snared an excellent one-handed catch.
Just as Kemar Roach achieved in the last Test in Trinidad, Shillingford was able to celebrate a 10-wicket haul in the match, becoming the the first West Indian spinner to achieve the feat since Lance Gibbs in 1966.
Roach then came on for his first bowl of the day and, coming around the wicket, beat Starc's bat and clipped the bails after the tail-ender had made 21
