Johnson's left-arm seamers claimed 6-31 to restrict Sri Lanka to a modest 191, with lower-order batsman Suraj Randiv top-scoring with a six and four boundaries in 41 runs.
Watson then hit six sixes and five boundaries to help Australia to its first win of the tour, assisted by half-centuries from captain Michael Clarke (53 not out) and former skipper Ricky Ponting (53) at Pallekele International Stadium.
Batting first after winning the toss, Sri Lanka made a steady start with Upul Tharanga and Tillakaratne Dilshan taking the total to 54 runs.
Tharanga, who was back in the side after serving a suspension for using a banned substance at this year's World Cup, hit seven boundaries for 34 runs before being bowled by left-arm spinner Xavier Doherty.
Offspinner David Hussey had Dilshan caught by Doug Bollinger for 29 with the total on 76.
Johnson entered the fray and had Mahela Jayawardene caught behind for 3 before running through the middle and lower order.
He also took the crucial wickets of Kumar Sangakkara (16), Dinesh Chandimal (15), Jeevan Mendis (2) and Randiv in eclipsing his previous best figures of 5-26 against India in 2007.
Australian openers Brad Haddin and Watson shared 59 runs before Haddin was bowled by spinner Ajantha Mendis for 12.
Watson was caught by Jayawardene off spinner Randiv after facing only 51 balls.
Clarke and Ponting put on 101 for the third wicket to ensure victory. Ponting hit five boundaries in 71 balls, while Clarke hit only one boundary in a measured knock.
The next ODI in the five-match series is on Sunday at Hambantota.
