The Foxes, who opened the scoring through Martyn Waghorn, remain at the foot of the Championship as goals from Andrew Crofts, Adam Drury and a Wes Hoolahan brace earned Norwich victory.
Matty Fryatt, who came on to score twice, was sent off for a goalmouth tussle with Leon Barnett after the Foxes' third goal, and that effectively killed off any hopes they had of taking a point.
Leicester, who were beaten 6-1 at Portsmouth last time out, got off to the perfect start, opening the scoring after just a minute and 20 seconds.
A superb solo run from Waghorn down the right side saw him beat four defenders before bursting into the box and slotting the ball past John Ruddy.
Norwich, who went into the game on the back of an inconsistent run, responded through Drury, but his shot was tipped around the post by Carl Ikeme.
Ikeme had to be at his best to keep out a cheeky lobbed effort from Andrew Crofts in the 12th minute, while the winger fired wide from 20 yards minutes later.
Crofts was looking lively, and after Russ Martin and Barnett had brought saves from Ikeme, he found the equaliser.
Grant Holt cut the ball back from the left for the Wales international and he drilled the ball low across the goalkeeper.
Leicester, who had managed just one Championship win before the start of play, almost regained the lead before the break but Steve Howard failed to get a decent connection on Tom Kennedy's cross.
Norwich had already gone close twice through Hoolahan and Simon Lappin at the start of the second half when they went ahead from the penalty spot.
Michael Morrison was adjudged to have handled a cross from Martin and Hoolahan stroked home the penalty.
It was all Norwich, with Holt heading wide from close in, Simeon Jackson firing in a shot which Ikeme tipped away and Elliott Ward sending a header past the upright.
Their pressure paid off in the 62nd minute when Hoolahan picked out Drury and he found the bottom corner to make it 3-1 on his 300th league appearance.
Leicester pulled a goal back three minutes later when Howard's shot was saved by Ruddy but substitute Fryatt was on hand to poke in the rebound for his first goal of the season.
The Canaries composed themselves and pushed forward again, going close through a Holt header from 12 yards.
They restored their two-goal advantage in the 74th minute when Holt laid the ball to Hoolahan and he smashed a half-volley into the top corner from 20 yards.
But Leicester refused to lie down and came back again to make it 4-3 when Fryatt scored a superb individual effort.
The striker was then shown a straight red card in his attempts to get the ball back, getting himself involved in a tussle with Barnett.
That allowed Norwich to relax in the closing stages, with Drury inches away from adding a fifth.