Opener Tillakaratne Dilshan and Kumar Sangakkara were at the crease on 39 and 44 respectively with Sri Lanka needing 201 more runs from the remaining 33 overs with nine wickets in hand at the Oval.
Kushal Perera was the only Sri Lankan batsman dismissed as he fell cheaply for six in the third over of the bowling of James Anderson with just 10 runs on the board.
Earlier, three England top-order batsmen hit half centuries before a late flourish by Ravi Bopara as the home side post a challenging 293 for seven.
The initial advantage seemed to have frittered away somewhat for England before Ravi Bopara played a cameo of 13-ball 33 not out to take the score near 300 mark. He struck 28 runs with the help of three sixes and two fours from the last over bowled by pacer Shaminda Eranga. England scored 92 runs from the last 10 overs.
Cook consumed 85-balls and hit just three fours while Trott found the boundary ropes five times in his 87-ball innings. The cautious approach was apparent from the very beginning with prolong periods without boundaries.
However, England somehow got the runs and kept the scoreboard ticking. Their ability to negotiate fast bowler Lasith Malinga (2/58), whose first spell of four overs yielded just 10 runs, without any damage saw them in a sound position by the halfway mark when the home side were 124 for one.