The Lambs made a bright start in front of their own fans, with midfielder John McGrath trying his luck from distance twice in the early exchanges.
The second drive forced a solid save from Canaries goalkeeper Paul Gallacher and soon after he should have been tested again, but Simon Heslop blazed over when well-placed on the right.
The Championship side hit back with Robert Earnshaw shooting over and Dickson Etuhu going close following a corner, but City were unable to extend Lambs keeper Jose Veiga.
However, the Conference strugglers were rocked by a double blow with half-time approaching as Norwich scored twice in three minutes.
First Darren Huckerby found space on the left and his cross to the near post was headed home from six yards out by the unmarked Dion Dublin.
Huckerby quickly added a second to give Peter Grant's side some breathing space, confidently firing past Veiga from 12 yards.
The game was over as a contest five minutes into the second half when Dublin flicked on a long throw and Lee Croft set up Huckerby to drive a low shot under Veiga.
Just past the hour mark Norwich made it 4-0 as Dublin seized on a defensive error to lob the ball in to the empty net past the stranded Veiga.
Tamworth got on the scoresheet themselves midway through the second half as Kyle Storer curled home an excellent right-foot shot from the edge of the area, but his side's hopes had long since disappeared.