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Chandimal made an unbeaten 206 in Sri Lanka's 554 all out, as the hosts managed a first innings lead of 190 in Galle.
Jayasuriya, a left-arm spinner, then took six wickets to help dismiss Australia for 151 in the final session on day four as the hosts bounced back from their opening loss to end the series at 1-1.
The tourists posted 364 with Steve Smith hitting an unbeaten 145 in the first innings but their batting faltered the second time around.
Off-spinner Ramesh Mendis trapped David Warner lbw for 24 before tea and Jayasuriya soon took over with his left-arm spin in the final session to send back Usman Khawaja (29) and Steve Smith (0) in one over to rattle the opposition batting.
The 30-year-old Jayasuriya, who claimed a six-for in the first innings, kept up the charge with another twin strike to expose the Australian tail.
Ramesh and debutant spinner Maheesh Theekshana took two each and Jayasuriya got the final wicket to trigger celebrations in the Sri Lankan camp and crowd.
Jayasuriya's match figures of 12-177 are the best ever by a Sri Lankan bowler on debut as he went one better than Praveen Jayawickrama, who took 11 wickets in his first Test last year.
But Chandimal set up victory after he launched an assault on the opposition bowlers with his final 47 runs coming from just 28 balls.
Chandimal, 32, smashed left-arm quick Mitchell Starc for a four and two sixes on consecutive balls to go past 200 as he leapt in joy amid raucous applause from the crowd.
Starc took four wickets while leg-spinner Mitchell Swepson took three to polish off the tail.
The hosts began the day on 431-6 in response to Australia's first innings 364.
Opening batsman Pathum Nissanka was ruled out before play started Monday with Covid-19, becoming the sixth Sri Lanka player to contract the virus during the past two weeks.
The overnight pair of Chandimal and Ramesh added 68 before Starc broke through.
Chandimal, who recorded his 13th Test hundred on Sunday, took on the opposition bowlers after losing Jayasuriya for nought as he went past his previous best of 164.
Chandimal and debutant Kamindu Mendis, who made 61, put together 133 runs for the fifth wicket to take the team into the lead on day three.
