The home team fought gallantly on the final day and almost snatched a highly unlikely draw before last man James Anderson spooned a Shaminda Eranga delivery to Rangana Herath at square leg, handing Sri Lanka their first series victory in England.
Anderson (duck) and Moeen Ali, who compiled a brilliant backs-to-the-wall 108 not out, defended valiantly as the last-wicket pair negotiated 20 nail-biting overs in the final session before succumbing off the penultimate ball of the day.
The real damage, though, for England was caused on Monday when they let slip a good position as Sri Lanka captain Angelo Mathews struck a magnificent 160 and paceman Dhammika Prasad took four wickets to reduce the hosts to 57 for five.
Defeat came on the back of a humiliating 5-0 Ashes rout in Australia at the start of the year but Cook, who has also struggled for runs and is now without a test century in 24 innings, said he would fight on.
“I’ve never quit on anything,” the skipper told reporters after his team were bowled out for 249 to lose by 100 runs. “I’m incredibly proud to be England captain.
“It’s a role you’re selected to do. If someone decides I’m not the right person for the job and the results don’t justify me doing it, then fine.
“But until that moment I’m desperate to try and help turnEnglish cricket around,” added opening batsman Cook.
“In every one of the 104 games I’ve played for England I’ve left everything out there. Sometimes I’ve done well and sometimes I haven’t. I’ve given it my all, all the time.”
Cook said he would take a few days off in the hope ofarresting his form slump before starting to prepare for the five-test series against India which begins next month.
“At the moment the runs are hard to come by,” he added. “Itdoes put more pressure on me and I’ve got to use these next 10 days well.
“I’ll rest up, get back to train incredibly hard and hopefully score some runs against India in the first couple of games.
“Nobody is guaranteed a place in this England team, you’veseen that with the young cricketers around now, they’re pushing for places. That’s the way it should always be.”
Sri Lanka batsman Kumar Sangakkara praised the efforts of man of the match Mathews after his captain turned the game around on Monday.
“Mathews showed he is a fantastic young man,” said Sangakkara. “The way he led us through the tour – through the best of times and some controversy – showed the strength of his character.”
Mathews was on the receiving end of much criticism after Sri Lanka ran out Jos Buttler while backing up at the non-striker’s end in the one-day series.
The visiting captain said he was overjoyed with the test series victory.
“It means everything,” explained Mathews. “We kept trying and didn’t lose our belief.
“Our bowlers are quite inexperienced but they tried to hit their lines and lengths and were brilliant.”
Coaching consultant Chris Adams, a former England test player, said: “It was a huge, huge win. Sri Lanka are great lads and I’m absolutely delighted for them”.
(Editing by Tony Jimenez)