The captain made a 57-ball 104 not out, smashing five sixes and 12 fours in a superb display of hard and clean hitting to help his team post a challenging 198-3 in the day-night match at the Pallekele International Stadium.
Australia were restricted to 163-8, with opener David Warner top-scoring with a 31-ball 53 with the help of two sixes and six fours.
Sri Lanka virtually wrapped up the match when they reduced Australia to 63-4 in the opening nine overs, with debutant off-spinner Dilruwan Perera doing the maximum damage with three wickets.
Dilshan needed 16 runs to complete his hundred in the last over bowled by paceman Mitchell Johnson [ Images ], but struck two sixes and as many fours. His second fifty came off just 16 deliveries.
The Sri Lankan captain and Jeevan Mendis (29 not out off 19 balls) plundered 104 runs in the last 7.3 overs to help their team set a stiff target.
"I am really satisfied. Earlier on I batted a bit slowly but then played big shots later. It was a good toss to lose," said Dilshan, who was named man of the match for his brilliant knock.
"Perera did a great job. Mendis came and took a few risks and when runs are coming from both ends, it is very difficult for the bowling side."
Dilshan, who also added 52 for the third wicket with Kumar Sangakkara [ Images ] (30), went for big shots after 15 overs, hitting four boundaries in an over from seamer John Hastings and then two sixes and as many fours off Shane Watson [ Images ].
"Full credit to Sri Lanka and Dilshan. We were in the game for 13-14 overs, but Dilshan took the game away from us," said Australia captain Cameron White [ Images ].
Sangakkara was batting confidently before driving Watson straight to White at mid-wicket. He hit four boundaries in his 22-ball knock.
Fast bowlers Brett Lee [ Images ], Johnson and Watson each took one wicket.
The second and final Twenty20 game will be played at the same venue on Monday.
Australia will also play five one-day internationals and three Tests on their tour.
M Jayawardene [ Images ] b Lee 11
T Dilshan [ Images ] not out 104
D Chandimal hit wkt b Johnson 11
K Sangakkara c White b Watson 30
J Mendis not out 29
Extras: (lb-7, w-6) 13
Total: (For 3 wkts in 20 overs) 198
Fall of wickets: 1-12, 2-42, 3-94
Bowling: Lee 4-0-38-1, O'Keefe 3-0-18-0, Johnson 4-0-47-1, Hastings 3-0-34-0, Smith 2-0-13-0, Watson 4-0-41-1.
S Watson b Kulasekara [ Images ] 10
D Warner c Dilshan b Kulasekara 53
S Marsh lbw b Perera 4
D Hussey b Perera 6
C White b Perera 7
S Smith st Sangakkara b Herath 12
B Haddin run out 7
M Johnson not out 22
S O'Keefe b Lakmal 22
J Hastings not out 14
Extras: (lb-1, w-5) 6
Total: (For 8 wkts in 20 overs) 163
Fall of wickets: 1-18, 2-39, 3-49, 4-63, 5-89, 6-103, 7-105, 8-144
Bowling: Kulasekara 4-0-39-2, Lakmal 3-0-23-1, Perera 4-0-26-3, Prasad 3-0-37-0, Dilshan 3-0-26-0, Herath 3-0-11-1.