England were 408 for eight at lunch on the second day, with Ali 85 not out -- his second-highest score in Tests after an unbeaten 108 against Sri Lanka at Headingley two years ago.
Now England hoped he could succeed where Alex Hales (83) and Joe Root (80) had failed Friday by going on to a century.
Together with Chris Woakes, Ali put on 92 for the seventh wicket, although both batsmen were dropped.
England's total was especially sizeable total given Sri Lanka had been dismissed for 91 and 119 during the hosts' innings and 88-run victory in the first Test of this three-match series at Headingley last week.
Left-hander Ali was 28 not out and Woakes, replacing the injured Ben Stokes, eight not out.
Sri Lanka held several brilliant catches on Friday, but Saturday's early overs saw them drop two chances.
Ali on 36 fended outside off stump against Nuwan Pradeep but Dimuth Karunaratne at a wide second slip couldn't hold the shoulder high chance as he moved to his left.
But that miss was as nothing compared to the reprieve given Woakes.
The Warwickshire all-rounder had yet to add to his overnight eight when he edged Shaminda Eranga straight behind.
Even allowing for the fact Chandimal was nursing a thumb injury suffered late Friday this was still an extraordinary miss by a Test match wicket-keeper.
Woakes cashed in, driving Eranga past point and hitting Suranga Lakmal straight back down the ground for well-struck fours.
The drinks break saw Kusal Mendis, one of several wicket-keeper/batsmen in the Sri Lanka squad, replace Chandimal behind the stumps.
Ali's forceful boundary off left-arm spinner Rangana Herath, his sixth four, then saw him to a 109-ball fifty.
Mendis had another dismissal when he caught Stuart Broad off Lakmal but by then England had already reached 400.
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