The Honduras international cancelled out Jack Compton's first-half opener early in the second period and after Ross McCormack had levelled for the home side following Bradford midfielder Michael Flynn's spectacular strike, Nunez converted his second from close range to decide a see-saw encounter.
Bradford, who had not won any of their last 13 encounters with Leeds and were chasing their first win over their arch-rivals since 1986, took the game to Simon Grayson's side from the outset and played their part in an absorbing derby before running out of steam.
Nunez, midfielder Lloyd Sam and defender Ben Parker were recalled by Leeds, while on-loan Tottenham goalkeeper Oscar Jansson made his debut for Bradford and defender Steve Williams returned to the side.
Nunez spurned several chances early on and Leeds were made to pay on the half-hour mark when Compton steered home a left-wing cross from former Falkirk team-mate Mark Stewart.
Bradford striker James Hanson headed narrowly wide and midfielder David Syers volleyed just the wrong side of the post as the visitors finished the first half in the ascendancy.
But City's lead lasted just 31 seconds into the second period. Sam fed Clayton after a mazy run and the latter's cross was slammed home from close range by Nunez.
Flynn regained the lead for Bradford in stunning style in the 57th minute when his shot on the turn from the edge of the area flashed into the top corner.
Leeds levelled again in a helter-skelter cup tie in the 70th minute as Clayton clipped another fine cross into the area and McCormack headed high into the net.
United then hit Bradford on the break for the winner with Sam's cross bouncing kindly for Nunez to convert from close range.