The tourists, set a history-defying target of 340, reached 111 for two at stumps on the third day on a dry tone that made batting progressively difficult.
The world's number one Test side lost openers Andrew Strauss and Alastair Cook cheaply, before Jonathan Trott and Kevin Pietersen repaired the damage by adding 63 for the unbroken third Wicket.
Trott was unbeaten on 40 and Pietersen was on 29, with England still needing 229 runs with eight wickets in hand to win the Test and take the initiative in the two-match series.
England, which slips to number two behind South Africa if they lose the series, never chased down such a big fourth innings target before in their Test history.
Their most successful chase is 332-7 against Australia at the Melbourne Cricket Ground as early as 1928. In Asia, there are 209-1 against Bangladesh in Dhaka in March, 2010.
England job has become more difficult because 253 is the highest fourth innings total made at the Galle International Stadium.
Cook, who scored 14, was given caught at the Wicket out Rangana Herath through the Decision Review System after on-field umpire Rod Tucker had negated the appeal.
Herath, whose left arm spin took him six wickets in the first innings, dismissed Strauss for 27 when the England captain failed to maintain on-disk down and was caught in a short mid-Wicket.
Pietersen was lucky still there at stumps after being abandoned by Kumar Sangakkara at backward short leg off Suraj Randiv when he was on 12.
Earlier, Sri Lanka were bowled out for 214 in their second innings in the post-lunch session with England off spinner Graeme Swann finishing with six for 82.
The hosts were reduced to 127-8 just before lunch, the lead of 252 runs, when Wicket-keeper Prasanna Jayawardene stepped in hitting an unbeaten 61 in the company of tailenders.
Jayawardene, who was caught off a Stuart Broad non-ball when on 29, put on 40 for the ninth Wicket with Chanaka Welegedara (13) and 47 to the 10th with Suranga Lakmal (13).
Swann bagged two of the three wickets that fell in the morning session after Sri Lanka resumed on Tuesday, the score of 84-5.
The night a couple of Dinesh Chandimal and Suraj Randiv took their sixth Wicket partnership to 42 before both batsmen fell in the space of one run.
Chandimal, who made 31, his talented Wicket for the second time in the match, when he attempted a shot off Monty Panesar and only managed to sky a catch to Kevin Pietersen at mid-off.
Randiv was declared leg-before by television official Bruce Oxenford after challenging on-field umpire Asad Rauf decision, giving Swann his fifth Wicket at the entrances.
Herath became Swann's sixth victim when he was bowled trying to sweep the off spinner.
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