Casson removed Dwayne Bravo for 69, and Clark got Shivnarine Chanderpaul for 50, as West Indies, chasing 475 runs to win, reached 316 for five in their second innings at the interval with Denesh Ramdin not out on zero and Jerome Taylor not out on seven on the fifth and final day of the Test at Kensington Oval.
For just over 90 minutes, Australia had been kept at bay by Chanderpaul and Bravo, after West Indies continued from their bedtime total of 235 for three.
The day started shakily for the Aussies, when Chanderpaul edged the first delivery from Clark through the slip cordon for a boundary.
Unfortunately for the Aussies, Chanderpaul and Bravo took this as a sign, and knuckled down to play circumspectly, and a few milestones fell by the way during this period.
First, Bravo stabbed a delivery from Casson into the square cover region for a single to reach his 50.
Then Chanderpaul reached 8,000 Test runs, when he glanced a delivery from Brett Lee to deep fine leg for a single.
Chanderpaul then tugged a full toss from Casson to mid-on for a single in the following over to reach his 50.
The Aussie bowlers would also suffer some punishment along the way, and an intriguing battle surfaced between Casson and Bravo which saw the young left-arm wrist spinner being carted three times over long-off for sixes.
But Australia captain Ricky Ponting persevered with Casson and he was rewarded, when Bravo was caught at silly point playing defensively forward to a delivery of no great merit to end a fourth-wicket stand of 122.
Ponting then claimed the second new ball after 83.1 overs, and Clark struck with the second delivery, when he had Chanderpaul adjudged lbw about 15 minutes before lunch.
Clark has taken two wickets for 40 runs from 18 overs, and Casson snared two for 80 off 21 overs.
Australia lead the three-Test series 1-0, after they won the opening Test at Sabina Park in the Jamaica capital of Kingston by 95 runs.
The second Test at the Vivian Richards Cricket Ground in the Antigua village of North Sound was drawn.
The Aussies have already retained the Frank Worrell Trophy - symbol of Test supremacy between the two sides.