The Foxes started the game brightly and only Palace 'keeper Speroni kept the game goalless early on with several fine saves. City came close to breaking the deadlock mid-way through the first half when Gallagher's pin point pass found Oakley who fired hard and low only to be denied by Speroni.
Leicester started the second half as dominantly as they ended the first and with 53 minutes on the clock, Martyn Waghorn's powerful shot was saved but Swiss full back Bruno Berner was well placed to score his second goal in three games to give the Foxes the lead.
A resolute City defence coped well with a late Palace charge and the team held on for a deserved three points that take them up to fourth place in the Championship table as other results went in favour of the Foxes.
Swansea were beaten at home by West Brom. Coventry and Blackpool also lost while Forest were shocked by Barnsley. City have an eight point play-off safety net now ahead of Sunday's visit of midland rivals Coventry.