Silva pounced from close range to put City ahead after an unconvincing start at the Etihad Stadium and Milner came off the bench to settle nerves in the last minute.
It was far from a fluent display from City but the result will at least take some of the heat off manager Manuel Pellegrini after a testing few days.
The hosts could justifiably claim victory was deserved after forcing veteran goalkeeper Mark Schwarzer to make several fine saves.
But the Foxes were denied three penalties - with two appeals looking particularly strong - and defeat was harsh on the hard-working bottom side.
Luck was therefore on Pellegrini's side after the build-up to the game had been dominated by criticism of his tactics in damaging losses to Barcelona and Liverpool and speculation over his future.
Source : PA
Source: PA