With the win, City cuts United's lead to two points in the league with four games left to play in the season. United takes on Aston Villa tomorrow at home and will need a victory to push their advantage back to five points.
Sergio Aguero grabbed two goals to back up Tevez and Adam Johnson chipped in one during stoppage time to complete the rout.
Norwich cut the lead to 2-1 in the 51st minute on a goal from Andrew Surman at Carrow Road. City put the hammer down after that goal and didn't let up, scoring four times in the final 40 minutes of play.