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The Sri Lankans had looked to be holding on for a draw until Siddle and Starc struck to secure Australia's first win of the home season, after losing to South Africa 1-0 in the recent three-Test series.
Starc claimed the last four wickets to end with five for 63, while Siddle was named man of the match after his four for 50 gave him nine wickets for the match.
Australia found it hard going to gouge out the batsmen on a fifth-day pitch until the wickets tumbled in the final two hours.
"We knew we had to be patient," said captain Michael Clarke. "No doubt I was confident that if we could bowl the balls in the right area for a long period of time that we would have success.
"It was going to be a day's work to get so many good batsmen out, but take nothing away from the bowlers.
"They had to execute their plans and their skills. Reverse swing was always going to be vital in this game and we've got three of the best reverse swing bowlers in the world in Sids (Siddle), Shane (Watson) and Starcy (Starc)."
Australia got the wicket of captain Mahela Jayawardene before lunch, Kumar Sangakkara in the middle session and the remaining six wickets in the final session after the tourists had appeared headed for a morale-boosting draw.
"Disappointed that we lost the game. It was a tough Test match in the sense that we were chasing the game probably after the third or fourth day but we kept fighting, especially in the first innings," Jayawardene said.
"Today we knew we had to bat through a couple of sessions sensibly, see what happens, and up until tea we were on target. And then two wickets after tea and that's when things probably went in the other direction."
Siddle struck in the fourth over after tea when Angelo Mathews needlessly flicked outside his off stump and nicked to Matthew Wade for 19, giving the Australians a glimmer of victory.
Siddle gave the Australians more hope when he trapped Samaraweera leg before wicket for 49 off 140 balls.
A few overs later Prasanna Jayawardene, who has scored four Test centuries, could not avoid a spitting delivery from Starc and gloved to Mike Hussey at second slip for 21.
Starc bowled a beautiful delivery to entice an edge off Nuwan Kulasekara who was caught behind for nine heading into the final 15 overs.
The irresistible left-armer bowled Rangana Herath for eight, leaving the Australians just one wicket from victory.
Starc finished up the innings when he had Shaminda Eranga caught behind for six to end Sri Lanka's dogged resistance.
The Australians had earlier removed key batsmen Mahela Jayawardene (19) and Sangakkara (63) but for a time were hitting a brick wall in their attempts to make further inroads.
At one stage Clarke even brought wicketkeeper Matthew Wade into the attack, with his lively medium pace utilised for a maiden final over before the tea adjournment.
The big-hearted Siddle made the crucial breakthrough in the middle session, trapping Sangakkara leg before wicket for 63 off 226 balls in the 76th over.
Australia removed Mahela Jayawardene midway through the morning session after Clarke, playing with an injured hamstring, bent low to take a sharp catch off Siddle in the 56th over.
Jayawardene had resisted for 77 balls for his 19 and stubbornly put on 65 runs for the third wicket with Sangakkara off 193 balls.
Sri Lanka have yet to win in Australia in 11 Tests.
The series now goes to Melbourne for the second Test starting on Boxing Day ahead of the final Sydney Test on January 3.
