Set 362 to win after Alex Hales had narrowly missed out on his maiden test hundred for England, who are 2-0 up in the series, Kaushal Silva and Dimuth Karunaratne took the visitors to 32 without loss by the close.
It leaves Monday's extended final day nicely poised with Sri Lanka facing the tallest of orders to score the runs that would secure a consolation victory.
After the morning session and part of the afternoon's play were lost to rain, England resumed on 109 for four and opener Hales was the picture of disappointment when he ended up falling six runs short of his first test ton.
With Alastair Cook, back in action after missing the start of the second innings with a bruised knee, also looking to push on towards a declaration at the other end, Hales was trapped lbw for 94 when trying to turn Angelo Mathews into the leg side.
He reviewed umpire Rod Tucker's decision but was left dismayed when the video replay showed the ball would have clipped the top of leg stump.
Cook, in the unusual position of batting at number seven, enjoyed himself, following up his first-innings 85 by helping himself to a six in an unbeaten 49.
Declaring on 233 for seven, Cook then gave his attack 12 overs to make a breakthrough but Silva and Karunaratne, who shared a hundred stand in the first innings, had few alarms against two of the world's top three pacemen, Stuart Broad and Jimmy Anderson.
(Reporting by Ian Chadband; Editing by Tony Jimenez)
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