Paul McVeigh did the damage to heighten relegation fears among the Leicester faithful, some of whom at the end of the match chanted for Levein's head.
Before this defeat the former Hearts boss admitted he was facing his toughest time in football as a player or manager and he will now be praying for a New Year revival after a dismal December which saw a return of just two points from a possible 18.
There was precious little goal action in the opening exchanges, the best of which saw Johannes Gudjonsson's long-range effort drift wide.
On 20 minutes McVeigh failed to control the ball with only Rab Douglas to beat after Darren Huckerby had whipped in a cross from the left.
Leicester hit back with Gareth Williams crashing a volley just wide before Ryan Smith squandered the home side's best chance when he was just wide of the target from close range after the visitors had failed to clear Richard Stearman's long throw.
Robert Green was forced into the first meaningful save of the game on 35 minutes when he was at full stretch to keep out a Paddy McCarthy's downward header.
After the break Norwich felt aggrieved when they failed to earn a penalty after Huckerby went down under a challenge from McCarthy.
But there was no debate on 71 minutes when the visitors took the lead with a fine goal.
Dickson Etuhu's clever through ball picked out McVeigh who outstripped the home defence before beating Douglas with a well struck shot from 16 yards.
Confidence now flowed through Norwich who pressed forward in search of a second goal to put the result beyond doubt and that almost arrived in the final minute when Huckerby squared the ball to Dean Ashton who crashed a 15-yard shot against the bar.