Right-arm seamer Kasprowicz took seven for 39 from 17.4 overs as Sri Lanka, chasing a daunting target of 312 in a low-scoring game, were bowled out for 162.
Mahela Jayawardene and Thilan Samaraweera put on 79 for the fourth wicket before the last seven batsmen tumbled for only 53 runs.
EARLY BREAKTHROUGHS
Sri Lanka lost their first wicket with the total on 23, captain Marvan Atapattu caught at slip by Shane Warne off Kasprowicz for 10.
Man of the match Glenn McGrath, who took five wickets in the first innings, kept the pressure on with two for 24 from 16 frugal overs.
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Going to lunch at 50 for three, Sri Lanka fought back through Jayawardene and Samaraweera before both fell in quick succession midway through the second session.
Samaraweera was caught behind off Kasprowicz for 32 and Jayawardene bowled by McGrath for 44.
Russel Arnold was out six minutes before tea, caught behind off Kasprowicz for 11 with the score 132 for six.
Kasprowicz struck four more times after tea, removing Tillakaratne Dilshan for 14, Upul Chandana for 17 and Nuwan Zoysa for one. Lasith Malinga was out next ball for a duck, caught behind to give Adam Gilchrist his fifth catch of the innings.
Scorecard:
Australia first innings 207 (Darren Lehmann 57; Chaminda Vaas 5-31)
Sri Lanka first innings 97 (Glenn McGrath 5-37)
Australia second innings 201 (Adam Gilchrist 80; Lehmann 51)
Sri Lanka second innings
M.Atapattu c Warne b Kasprowicz 10
S.Jayasuriya lbw b McGrath 16
K.Sangakkara run out 0
M.Jayawardene b McGrath 44
T.Samaraweera c Gilchrist b Kasprowicz 32
T.Dilshan c Gilchrist b Kasprowicz 14
R.Arnold c Gilchrist b Kasprowicz 11
U.Chandana b Kasprowicz 17
C.Vaas not out 10
N.Zoysa c Gilchrist b Kasprowicz 1
L.Malinga c Gilchrist b Kasprowicz 0
Extras (nb-4, w-2, lb-1) 7
Total (all out, 65.4 overs) 162
Fall of wickets: 1-23, 2-23, 3-30, 4-109, 5-113, 6-132, 7-141, 8-152, 9-162
Bowling: McGrath 16-9-24-2 (w-1, nb-1), Gillespie 13-2-37-0 (w-1), Kasprowicz 17.4-3-39-7 (nb-3), Warne 19-2-61-0
Man of the match: Glenn McGrath (Australia)
The second Test of the two-match series in Cairns begins on July 9.