The Australians claimed six wickets in the final session to take a 1-0 lead in the series with two Tests to play in Melbourne and Sydney.
Pace bowlers Peter Siddle and Mitchell Starc bowled Australia to victory after it appeared that Sri Lanka may hold on for a gutsy draw.
Starc claimed five wickets for 63 and Siddle captured four for 50 to finish with nine wickets for the match.
Kumar Sangakkara top scored with 63 off 226 balls.
The teams head to Melbourne for the second Test starting on Boxing Day with the final Test in Sydney on January 3.
Match scores: Australia 450 for 5 declared and 278; Sri Lanka 336 and 255.
Australia 1st innings 450 for 5 declared (M. Hussey 115 not out, P. Hughes 86; C. Welegedara 3-130)
Sri Lanka 1st innings 336 (T. Dilshan 147, A. Mathews 75; P. Siddle 5-54)
Australia 2nd innings 278 for 9 (D. Warner 68, M. Clarke 57 retired hurt, E. Cowan 56; R. Herath 5-95)
Sri Lanka 2nd innings (overnight 65 for 2)
D. Karunaratne b Starc 30
T. Dilshan c Wade b Watson 11
K. Sangakkara lbw b Siddle 63
M. Jayawardene c Clarke b Siddle 19
T. Samaraweera lbw b Siddle 49
A. Mathews c Wade b Siddle 19
P. Jayawardene c Hussey b Starc 21
N. Kulasekara c Wade b Starc 9
R. Herath b Starc 8
S. Eranga c Wade b Starc 6
C. Welegedara not out 0
Extras (b10, lb8, w1, nb1) 20
Total (all out, 119.2 overs) 255
Fall of wickets: 1-26 (Dilshan), 2-47 (Karunaratne), 3-112 (M. Jayawardene), 4-151 (Sangakkara), 5-201 (Mathews), 6-218 (Samaraweera), 7-235 (P. Jayawardene), 8-247 (Kulasekara), 9-250 (Herath), 10-255 (Eranga)
Bowling: Starc 28.2-7-63-5 (1nb, 1w), Siddle 26-11-50-4, Watson 27-6-54-1, Lyon 32-12-57-0, Hussey 1-0-5-0, Warner 4-0-8-0, Wade 1-1-0-0
Result: Australia won by 137 runs
Australia lead series 1-0
Man-of-the-match: Peter Siddle (AUS)
Umpires: Nigel Llong (ENG), Tony Hill (NZL)
TV umpire: Aleem Dar (PAK)
Match referee: Chris Broad (ENG)