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It had looked like ending goalless until Dickov's strike as neither side had looked capable of forcing a breakthrough.
Andy Johnson squandered two early chances for the visitors while City keeper Ian Walker showed his class with a stunning save to keep out a close-range effort from Hayden Mullins.
Leicester in contrast did not fashion a clear chance until the hour, when skipper Matt Elliott twice sent headers narrowly wide of the target, and the unmarked James Scowcroft headed straight at Cedric Berthelin.
Tommy Black then blew a great chance to put Palace in front when he ran in on goal unchallenged from 40 yards out, only to see a weak shot parried by Walker.
And with just 15 minutes remaining Dickov pounced for his 15th goal of the season to snatch all three points.
Muzzy Izzet carved open the Palace defence for the first time all afternoon but Leicester were frustrated to see Berthelin save bravely at the feet of Scowcroft.
The ball rebounded back to Izzet and his clever touch gave Dickov a yard of space to curl home a glorious strike from 20 yards.
Palace threw on former Foxes player Ade Akinbiyi in a frantic chase for an equaliser that never looked like coming.
Leicester now seem certain to claim an automatic promotion slot - but if they do go up they will have to play much better than they did today in order to survive.