Burke put the Blues ahead on 27 minutes and his side were able to withstand considerable pressure from the hosts to hold on for the win.
Birmingham fell to a 5-0 home defeat at the hands of Barnsley a week ago, but their defence, bolstered by the addition of new signing Paul Robinson, managed to keep the Seagulls out at the Amex.
Brighton will feel they should have got something from the match after dominating for much of the second half, but were unable to unlock the visitors' defence and were denied their sixth straight league win.
Both teams struggled to get a foothold in the match in the early stages, but a 20-yard left-footed volley from debutant Stephen Dobbie gave the encounter a much-needed injection of life after 23 minutes - his effort drifting wide of the mark.
However, it was Burke who broke the deadlock for Birmingham in the 27th minute with a superb 25-yard volley into the top-left corner of the goal.
Leroy Lita could have doubled the Blues' lead almost instantly, but the striker, on loan from Swansea, scuffed his shot and goalkeeper Tomasz Kuszczak collected.
Brighton continued to look bright for the remainder of the half, despite going behind, and a dangerous Buckley ball across the six-yard box in the 36th minute was unable to find a team-mate.
Buckley looked a threat throughout the first half, but will have been ruing his judgement of Birmingham's defensive line as he was punished for being offside on four occasions in the first 45 minutes.
The Amex Stadium descended into darkness at half-time as a power cut forced the players to return to the changing rooms after emerging for the start of the second half.
Normal service was resumed after a 25-minute delay and it was Brighton who carved out the early chances following the extended break.
Dobbie went close with a headed effort from a Wayne Bridge cross on 51 minutes, before the Scot had a 25-yard strike saved by Blues goalkeeper Jack Butland.
At the other end, Kuszczak had little to do for much of the second half as the hosts subjected Birmingham to lengthy spells of sustained pressure.
Wade Elliott became the first player to enter the referee's book in the 66th minute after a foul on Bridge, while Gordon Greer earned Brighton's only booking for dissent seven minutes from time.
Brighton kept pressing for an equaliser, but the visitors' defence held firm to keep the Seagulls at bay and come away with a much-needed victory.