Their failure to break the deadlock was punished in the 18th minute when Tevez sent a swerving effort past John Ruddy in the Norwich goal from a narrow angle.
Tevez played a role in City's second of the afternoon nine minutes later, flicking the ball into path of fellow Argentine Sergio Aguero, who sent the ball into the top corner of the net from 18 yards out.
Norwich manager Paul Lambert made two substitutions at the break and one of them, Andrew Surman, reduced the arrears in the 51st minute when he capitalised on a mistake from City goalkeeper Joe Hart to score into an empty net.
However, three goals in the space of seven minutes ended any hopes a Norwich comeback. First Tevez made it 3-1 in the 73rd minute when he headed over Ruddy, before Aguero curled an effort past the former Everton goalkeeper.
Tevez completed his trio of goals in the 80th minute, rounding Ruddy and scoring into an empty net following an error by Ryan Bennett.
Man United play their game in hand tomorrow afternoon against Aston Villa.