Root (104 no) and James Taylor (68) made a mockery of the consensus that England would struggle to chase 240 on an awkward, spinning surface at the Pallekele International Stadium.
Instead, after the loss of both openers, the third-wicket pair put on 104 together and then Root stayed untilthe end, first in another important stand with Ravi Bopara and then to ensure a minor late wobble did not cost his team.
The win helped England reduce the series deficit to 3-2 with two more matches to play, and captain Cook compared Root’s batsmanship to the “masterclass” he had witnessed from Sri Lanka great Kumar Sangakkara 24 hours earlier.
Sangakkara made 91 out of 239 all out, eventually falling as one of Chris Woakes’s six victims, before rain intervened between innings and left England with what seemed likely to be a tough task on the reserve day.
Thanks principally to Root and Taylor, adding another half-century to his 90 in Colombo, Cook had plenty to smile about.
“The way we bowled, first of all, was fantastic,” he said. “It’s great credit to Chris Woakes that he’s got two of the best three sets of figures (for England in ODIs).
“Then the way James Taylor, and particularly Rooty, played was just phenomenal batting today.
“I thought the way Sanga played yesterday was a masterclass in batting, and I thought Rooty hit the same level as that.
“Sanga was brilliant, and when Rooty went on and was there at the end that was in the same league.”
The upshot is that England still have a chance of starting their World Cup winter with a much-needed series victory – in difficult climes.
“This is a really important win today,” said Cook. “Obviously, Colombo hasn’t been that kind to us, but we’ve won away from there. It would be great to get to Colombo 3-3 and have a series final.”
Match report: Page 24.