Sri Lanka edge Bangladesh by two wickets to reach Asia Cup Super Four
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Chasing 184 for victory in a winner-takes-all game, Sri Lanka rode on a key 54-run stand between Mendis (60) and Shanaka (45) before the tail helped achieve their target with four balls to spare in Dubai.
Mendis fell after his 37-ball knock and Shanaka was out in the 18th over.
But Chamika Karunaratne, with a 10-ball 16, and Asitha Fernando, unbeaten on 10, held their nerve to knock Bangladesh out of the tournament, which acts as a tune-up to the T20 World Cup in October-November.
Needing eight of the final over, Fernando hit a four and then took two runs off a no ball to trigger celebrations in the Sri Lankan camp.
Bangladesh fast bowler Ebadot Hossain claimed three wickets but gave away 51 runs in his four overs.
Ebadot broke through in his first over to send back Pathum Nissanka for 20 and three balls later took the wicket of Charith Asalanka for one.
Mendis, who was dropped on two by wicketkeeper Mushfiqur Rahim, survived a reprieve on 29 when he was given caught behind but the delivery from off-spinner Mahedi Hasan turned out to be a no ball.
Eabadot dismissed Danushka Gunathilaka as Sri Lanka slipped to 67-3 and soon 77-4 with the departure of the big-hitting Bhanuka Rajapaksa.
Mendis was then kept company by Shanaka as the two put the chase back on track and soon took the attack to the opposition bowlers.
Shanaka hammered Ebadot in a 22-run 13th over while Mendis reached his fifty in 31 balls.
Bangladesh spearhead Mustafizur Rahman sent the Tigers fans wild with the wicket of Mendis, who hit four fours and three sixes in his knock.
Earlier, put in to bat first, Bangladesh posted 183-7 after a 57-run fifth-wicket partnership between Afif Hossain (39) and Mahmudullah Riyad (27).
Mosaddek Hossain chipped in with a useful 24 not out off nine balls to finish the innings with a flourish.
Sri Lanka took regular wickets, with Wanindu Hasaranga and Karunaratne claiming two each to help the island nation move into the next stage alongside Afghanistan from Group B.
The winners of the Group A game between Pakistan and Hong Kong on Friday will secure a Super Four spot alongside India.
Sri Lanka will face Afghanistan in the opening Super Four clash in Sharjah on Saturday.