In an encounter between the npower Championship's two most in-form teams a single goal was always likely to settle matters, and so it proved as the 6ft 8in Serbian - troubled by groin and hip injuries since the club's Wembley triumph over Arsenal - fired home after 35 minutes.
Blues goalkeeper Boaz Myhill had earlier kept Blues on level terms with an impressive save to keep out Adam Clayton's vicious volley before again coming to the hosts' rescue late on to tip over Ross McCormack's long-range effort.
Zigic and substitute Chris Wood brought good saves from Paul Rachubka, otherwise there was little serious threat to the scoreline amid what remained an intriguing and hard-fought clash.
Defeat ended fifth-placed Leeds' seven-match unbeaten league record and moved Blues to within a point of them. Wins in their two games in hand would take the Blues into second.
Leeds enjoyed the first real opening of the encounter after 11 minutes when fit-again Scotland international Robert Snodgrass fired across goal and wide from 15 yards. Blues were indebted to a genuine save of the season contender from Myhill after 24 minutes, the on-loan West Brom goalkeeper somehow palming Clayton's volley over the crossbar.
And it was proved all the more crucial shortly after the hour mark as Blues took the lead through Zigic. Stephen Carr broke down the right and delivered a cross which was not cleared by the Leeds defence and the ball found its way to the towering marksman who controlled and shot, seeing the ball take a deflection off Paul Connolly before beating Rachubka.
Zigic was denied what would have been a crucial second on the stroke of half-time as Rachubka used his feet to block well. Chris Burke fired narrowly wide for Blues after the restart before Snodgrass drilled over from distance.
As the second half wore on it was Leeds who were again forcing the issue as they chased their first point of any kind at St Andrew's since 1988, although Myhill remained largely untested. Having weathered the pressure Blues briefly took the game to Leeds, but it was the visitors who soon turned the screw once more as the clash entered the closing stages.
Myhill turned over McCormack's strike from distance in the 86th minute before Wood saw a shot of his own parried over the crossbar by Rachubka in injury-time, leaving Blues to maintain their unbeaten home record in the league at St Andrew's, where they have still conceded just one goal this term.