A triumphant cry on clinching record (5)
I believe the answer is:
alpha
'a' is the definition.
(the Greek letter A)
'triumphant cry on clinching record' is the wordplay.
'triumphant cry' becomes 'a-ha' (you might cry 'aha!' in triumph).
'on clinching' is an insertion indicator (inserted letters are secured or clinched).
'record' becomes 'LP' (abbreviation for long play record).
'aha' enclosing 'lp' is 'ALPHA'.
(Other definitions for alpha that I've seen before include "First-rate" , "Burrowing American rodent" , "Greek vowel" , "Foreign character" , "A for Aristophanes!" .)