After rain forced play to be suspended on Wednesday, the visitors returned to chase down a 240-run victory target with five balls to spare. They trail the hosts 3-2 in the series.
The 23-year-old Root (104 not out) added 104 for the third wicket with James Taylor (68) after England lost both openers cheaply to off-spinner Sachithra Senanayake.
Senanayake, back in the Sri Lanka team after being cleared to bowl again with a remodelled action, bowled Moeen Ali (four) in the second over.
He then compounded England's problems by getting under-fire captain Alastair Cook, who has faced criticism following a string of poor results and lack of runs, out lbw for 20.
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