Left-arm spinner Doherty, who was named man of the match, finished with a career-best 4-28 and paceman Lee with 4-15 as Australia dismissed Sri Lanka for 132 before achieving the modest target with 22 overs to spare.
Opener Shaun Marsh (70) was Australia's batting star as he put on 97 for the third wicket with skipper Michael Clarke (38 not out) after paceman Lasith Malinga removed Shane Watson and Ricky Ponting in the same over.
"It was a great performance with the ball," said Clarke.
"Doherty was fantastic and Lee took four wickets as well. Marsh won the game pretty much on his own. Another game is coming up and we would like to finish 4-1."
Debutant leg-spinner Seekkuge Prasanna dismissed Marsh and the Hussey brothers, Michael and David, in the space of four deliveries, but Australia needed just 10 runs to win with five wickets in hand at that stage.
Marsh, playing his first match of the series, hit one six and 11 fours in his impressive 80-ball knock.
Sri Lanka were earlier 95-3 following a 71-run stand for the third wicket between Mahela Jayawardene (53) and Kumar Sangakkara (31), but lost their last seven wickets for 37 runs in a dramatic collapse.
Skipper Tillakaratne Dilshan (12) was the other batsman to reach double figures in Sri Lanka's dismal performance on a slow wicket.
"We bowled well but cannot ask bowlers to defend 130 against a strong batting line-up. I think Prasanna has a good future," said Dilshan.
Jayawardene, dropped on 34 by wicket-keeper Brad Haddin off paceman Mitchell Johnson, hit just four boundaries in his patient 102-ball knock before falling to Lee in the 39th over.
Lee and Doug Bollinger did not allow Sri Lanka to make a solid start after the hosts elected to bat as they struck in their opening spells, removing Dilshan and Upul Tharanga (eight).
Sangakkara, hit on the helmet by a Bollinger bouncer off the first ball he faced, steadied the innings with Jayawardene before Doherty bagged three wickets in two overs.
The Australian spinner broke the partnership when he dismissed Sangakkara, caught by Lee at long-on while attempting a big shot. He then trapped Chamara Silva leg-before in the same over.
Doherty got his third wicket when he had Angelo Mathews stumped for six runs in his next over before accounting for Prasanna.