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.
The Yorkshire right-hander, who hit seven fours in his 117-ball knock, reached his hundred by slog-sweeping Ajantha Mendis for a six in the penultimate over.
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.
Taylor and Root consolidated initially before launching into the bowlers and laying down a solid platform for the chase.
The 24-year-old Taylor, who hit five fours and a six, showed nimble footwork against the spinners to get to his second successive half-century before he was dismissed by Thisara Perera.
Root and Ravi Bopara (28) took England close to the target with a fourth-wicket stand of 86 runs. Bopara and Eoin Morgan (five) fell in quick succession with England in touching distance of the win.
England paceman Chris Woakes (6-47) tore through the Sri Lankan lower order to bowl out the hosts for 239 in 49 overs on Wednesday despite a cultured 91 from Kumar Sangakkara.
A torrential downpour then kept the players off the pitch and prompted the match officials to abandon the day's play.
(Writing by Sudipto Ganguly in Mumbai; editing by Ed Osmond)
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