The Canaries had gone in front after just 17 seconds through Chris Martin, but lost three players to injury before the hour-mark as Jason Puncheon and Leven turned the game on its head to consign Paul Lambert to his first league defeat as City boss.
City made a dream start when Dean Lewington's rash clearance found Martin, who cut in from the right before stroking the ball past Willy Gueret from the edge of the box.
But their bright start was undermined when Stephen Hughes was carried off and left-back Adam Drury limped off in the opening 25 minutes.
Jabo Ibehre and Stephen Gleeson went close for the home side before Puncheon curled in a stunning 20-yard free-kick after 57 minutes for his first goal of the season.
Norwich's injury misery continued with centre-back Michael Nelson being replaced, before Martin was adjudged to have fouled Lewis Gobern in the box and Leven coolly converted from the spot.