An Asian city managed to continue (7)
I believe the answer is:
rangoon
'an asian city' is the definition.
(I've seen this before)
'managed to continue' is the wordplay.
'managed' becomes 'ran' (both can mean organised or was in charge of).
'to continue' becomes 'go on' ('go on' can be a synonym of 'continue').
'ran'+'goon'='RANGOON'
(Other definitions for rangoon that I've seen before include "Former Burmese capital" , "Former name of the capital of Myanmar" , "old Asian city" , "Former capital of Burma" , "It's now Yangon in Myanmar" .)