Broad finished with 3-36 and left-arm seamer Sidebottom with 2-41 as Sri Lanka struggled to score freely on a slow pitch after winning the toss in the day/night match.
Chamara Silva top-scored with a fighting 73 for his second successive half-century, but received little support from the other end.
He hit nine fours in his 109-ball knock.
England have already gained a winning 3-1 lead in the series.
Sri Lanka were strongly placed at 74-1 before losing their way against England’s disciplined pace-spin combination.
Left-arm spinner Monty Panesar, who came in place of the injured Graeme Swann, conceded just 31 runs in his 10 overs while part-time spinner Owais Shah grabbed a vital wicket in the middle overs.
Sri Lanka put on 45 for the opening wicket in only seven overs, with Sanath Jayasuriya hitting two sixes in his 18-ball 26 and debutant Dilruwan Perera playing some attractive shots in his 41-ball 30.
But Sidebottom ended Sri Lanka’s early flourish with two wickets in an over, removing Perera and then skipper Mahela Jayawardene for a duck.
Kumar Sangakkara (26) also failed to capitalise on the start. He looked like steadying the innings with Silva before off-spinner Shah struck.
Sangakkara stood at the wicket even after being bowled round the legs as South African umpire Rudi Koertzen, apparently unsighted by the bowler, had initially called it a wide ball.
The batsman had to leave when third-umpire Gamini Silva ruled after watching the TV replays that the ball hit the leg-stump.
Silva, who scored 67 in the previous game, kept one end intact before becoming the ninth man out in the 47th over.
Scoreboard
SRI LANKA:
M.K.D. Perera c Mustard b Sidebottom 30
S.T. Jayasuriya c Pietersen b Anderson 26
K.C. Sangakkara b Shah 26
M. Jayawardene c Broad b Sidebottom 0
L.P.C. Silva c Anderson b Broad 73
T.M. Dilshan run out 9
J. Mubarak c Bopara b Panesar 6
K.S. Lokuarachchi lbw b Bopara 0
W.P.U.J.C. Vaas b Broad 15
C.R.D. Fernando b Broad 12
S.L. Malinga not out 0
EXTRAS: (LB-4, W-9, NB-1) 14
TOTAL: (all out, 48.1 overs) 211
FALL OF WKTS: 1-45, 2-74, 3-74, 4-108, 5-131, 6-158, 7-159, 8-193, 9-210.
BOWLING: Anderson 8-1-48-1; Sidebottom 10-2-41-2 (3w); Broad 9.1-0-36-3 (1nb, 2w); Panesar 10-0-31-1; Shah 3-0-18-1 (1w); Bopara 8-1-33-1 (1w).
ENGLAND: A.N. Cook, P. Mustard, I.R. Bell, K.P. Pietersen, P.D. Collingwood, O.A. Shah, R.S. Bopara, S.C.J. Broad, R.J. Sidebottom, M.S. Panesar, J.M. Anderson.
UMPIRES: E.A.R. de Silva (Sri Lanka) and R.E. Koertzen (South Africa).
TV UMPIRE: M.G. Silva (Sri Lanka).
MATCH REFEREE: J. Srinath (India).—AFP