Australia ended day three at 200-6, extended the overall lead to 310 runs. Earlier West Indies were bundled out for 218 in their first dig.
Australia are leading the series 1-0 after winning the series-opener in Barbados. The second Test at Port of Spain was drawn due to rain.
An 87-run partnership between Ponting (57) and Cowan (55) rescued Australia from a rocky start but failed to sustain momentum and lost three wickets for 32 runs.
David Warner (11) and Shane Watson (5) got the hosts off to a worst start and perished to paceman Roach and spinner Shane Shillingford respectively to leave visitors 25-2.
Ponting and Cowan negotiated with both pace and spin attack with a levelheaded approach, concentrating on singles and doubles and propelling the loose deliveries past rope.
Cowan, who hit five boundaries in his 123-ball knock, fell to spinner Narsingh Deonarine caught by skipper Darren Sammy at covers but dauntless Ponting carried on the acceleration of flow of runs on board in the company of his successor Michael Clarke (25).
Ponting and Clarke added 56 runs before Roach, who has troubled Ponting so often removed him with a short, rising delivery that was top-edged to Chanderpaul.
Shillingford, the first Dominican to play on his own backyard, combined with Darren Bravo two overs later to get rid of Clarke.
Australia were not recovered from the two simultaneous dismissals yet before Deonarine struck again to snare Matthew Wade leg-before.
Mike Hussey (17) and Ryan Harris (4) took Australia to stumps without any further loss.
Earlier, Resuming the innings at 165-8, the hosts were finally bowled out for 218 half-hour before lunch.
Chanderpaul (68) shared a 66-run ninth-wicket stand with Ravi Rampaul (31) and 32-run last wicket stand with Roach (9 not out) before being adjudged lbw off Mitchell Starc.
Cowan was confident that 350 plus would be a defendable lead and they can overwhelm the hosts under it.
“I think we’re in control, definitely. I think 300 already is a big score. If we can get to lunch unscathed and maybe have 350, 360 lead, all of a sudden I think that’s a winning total,” he told reporters after the close of play.
“We’re confident we’ve got heaps of runs on the board already.”
West Indies VS Australia 3rd Test Scorecard
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