Roberto Mancini’s side moved within two points of title rivals United thanks to a treble from Argentina forward Tevez, two goals from Sergio Aguero and an Adam Johnson strike.
United now need to beat Aston Villa at Old Trafford on Sunday to re-establish their five-point advantage at the top.
After City’s defeat at Arsenal last weekend, it was widely expected that United would cruise to their 20th English league crown, but the champions’ midweek stumble at Wigan and back-to-back wins featuring 10 goals for Mancini’s men has breathed new life into the title race.
Norwich were swept aside before half-time as Tevez opened the scoring with swerving shot from long-range that deceived goalkeeper John Ruddy in the 18th minute.
Tevez was instrumental in the second goal nine minutes later. Aguero played a pass into Tevez and the Argentine’s deft back-heel allowed his compatriot to fire home from the edge of the area for his 27th goal of the season.
Norwich midfielder Andrew Surman reduced the deficit in the 51st minute, but Tevez poked home in the 73rd minute and Aguero netted a brilliant solo effort two minutes later.
Tevez completed his treble in the 80th minute and Johnson clinched City’s first away league win since February with a stoppage-time effort.