The pace duo built on man-of-the-match Owais Shah's superb 82 as England dismissed the World Cup runners-up for 169 after posting 234-8 to win their first one-day international in Sri Lanka since 1982.
The victory also helped England level the five-match series 1-1, having lost the first one-dayer by 119 runs at this venue on Monday.
The third game will be played here on Sunday.
Skipper Paul Collingwood (42 and 2-31) and Graeme Swann (34 and 2-27) also caught the eye with their all-round performances.
"It was very disappointing to lose the first game.
The boys showed a lot of character to bounce back. We were fantastic in all departments of the game," said Collingwood.
"Owais played a magnificent innings, to get such a score on a pitch like this and play the shots.
It was a good toss to win. We then took early wickets and put them under pressure.
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Left-arm paceman Sidebottom (2-23) and Broad (2-42) troubled the Sri Lankan batsmen with pace and movement under lights to reduce