Extremely elegant primarily in cap -- and tie! (4,4)
I believe the answer is:
dead heat
'tie' is the definition.
(I know that dead heat is a type of tie)
'extremely elegant primarily in cap' is the wordplay.
'extremely' becomes 'dead' (as in a phrase like 'dead easy').
'primarily' indicates taking the first letters.
'in' means one lot of letters goes inside another.
'cap' becomes 'hat' (cap can mean a hat).
The initial letter of 'elegant' is 'e'.
'e' placed inside 'hat' is 'heat'.
'dead'+'heat'='DEAD-HEAT'
'and' is the link.
(Other definitions for dead heat that I've seen before include "A draw in race, competitors finish exactly even" , "sporting tie" , "Tie for first place in race" , "Race in which no winner is apparent" , "Exactly tied race" .)