Spinner Graeme Swann struck with just his second ball in the penultimate over before the break in Perth to remove Clarke in his 100th Test.
With centuries in both the opening two Tests won by Australia, Clarke had looked in impeccable touch as he cruised to 24, but he advanced to Swann and hit the ball straight to English counterpart Alastair Cook, also playing his 100th Test, at short mid-wicket.
At lunch, Australia were 107-3 with opener David Warner again in good touch and not out 49, while Steve Smith was yet to score.
Warner was dropped on 37, when the recalled Tim Bresnan could not hang on to a sharp caught and bowled chance, and the left-hander responded by lofting the England seamer down the ground for six from the last ball of the over.
The opener pushed the ball to mid-on and set off for the quick single, but Anderson gathered and hit the stumps at the bowler's end with the Australian just short of his ground with only 13 runs on the board.
It was an important scalp for England, with the ball barely beating the bat in the opening hour and both Warner and Shane Watson looking in excellent touch as the score raced past 50 in the 10th over.
Australia named an unchanged line-up with pace bowler Ryan Harris selected despite struggling with a sore knee in the lead-up.
The tourists made just one change from the team beaten in the second Test in Adelaide, with seamer Bresnan returning from an injury lay-off at the expense of spinner Monty Panesar.
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