Alfonso Pedraza struck nine minutes from time with a neat finish to complete the victory over mid-table Birmingham.
"I thought we were second best for the first 60 minutes," Leeds manager Garry Monk told Sky Sports.
"We had a lot of defending to do, but from that point on we took the front foot and got good goals at good times."
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Leeds, bidding to return to the Premier League for the first time since 2004, are seven points behind second-placed Brighton & Hove Albion who occupy the second automatic promotion place.
Newcastle United top the standings on 73 points ahead of Brighton with 71, both from 34 games, Huddersfield on 65 from 33 and Leeds who have 64 from 35 matches.
(Reporting by Ed Osmond; Editing by Ken Ferris)
