League Two side Dagenham began the more positive and raced into an eighth-minute lead when Mark Arber found Sam Saunders, whose low cross was turned in from close range by Matt Ritchie.
But the Foxes hit back and levelled a minute later when Fryatt received a return pass from Nicky Adams to score from eight yards.
The Daggers continued to press and from a quick break in the 25th minute edged back in front. David Martin had beaten away Ritchie's fierce shot and Ben Strevens pounced to sweep home the loose ball.
But the visitors were pegged back again in the 28th minute when Magnus Okuonghae handled inside the box and Fryatt converted from the penalty spot.
City found themselves in front for the first time in the 55th minute when Marc Edworthy picked out Fryatt, who completed his hat-trick with a superb right-footed chip over the advancing Tony Roberts in the Dagenham goal.