The out-of-favour striker made the most of his first start of the season, finding the net twice inside four second-half minutes to add to Demarai Gray's opener at Bramall Lane.
Caolan Lavery pulled one back for the Championship outfit late on but, rather harshly on the balance of things, Ahmed Musa rifled home a fourth at the death to end United's seven-month winning run on home soil.
In the first meeting between these two teams in this competition since 1969, they largely cancelled each other out in a fairly lively first half.
Goalscorers Musa and Slimani, two of a combined 16 changes made by both sides, linked up 13 minutes in for the former to fire into the side-netting from a promising position.
The hosts' best chance of the opening 45 minutes fell the way of Ched Evans who, on his first start for the club in five years after rejoining in the summer, could not quite loop his headed attempt over Hamer following a smart Lavery knockdown.
Lavery had a go himself before the half was out, heading over from a left-sided cross, while David Brooks's effort from 25 yards out very nearly crept under the visiting keeper.
Lavery had a chance to open the scoring early in the second half after Christian Fuchs failed to clear his lines, which the Northern Irishman could not truly capitalise on as he only picked out Hamer's gloves.
A first goal since February for Gray gave Leicester the lead 52 minutes in, however, as a swift break culminated in the wideman taking it past two players and then tucking through the legs of Eastwood.
The Blades had a half-decent opportunity to level the match, with Brooks yet again sending a shot down the middle for Hamer to routinely stop.
Then came a killer second for the three-time winners, converted by Slimani after latching on to Andy King's through-ball and rounding Eastwood for a simple finish into the empty net.
Slimani did not have to wait long for a second, converting from a Gray cross at the back post moments after the English winger was denied at full stretch by a busy Eastwood.
There was still time for a hat-trick goal, wrongly ruled out by the officials after Wilfred Ndidi was deemed to be in an offside position in the build-up.
Lavery did pull one back eight minutes from time, catching Hamer out at his front post with what appeared to be a cross from the right, but Musa battled through the opposition defence and picked out the roof of the net from the final act of the match for his side's fourth.