The start to the game was end to end but neither side created many clear-cut opportunities until Zoe Cossey stung the palms of City keeper' Hope Armstrong.
Moments later, the Canaries took the lead on the break as Garrett capitalised on a long ball over the top and burst through on goal before beating Sian Fagg in the Town net.
Town nearly responded shortly after but Cossey was again denied by Armstrong before Natalie Richardson also forced the Norwich stopper into a smart low save.
However, the Canaries had a glorious chance to go into the break 2-0 up but City skipper Kate Parsons saw her glancing header drop just wide of the post.
Sophie Welton then had a great opportunity to level the game up before the break but the winger fired just wide across the face of goal after a terrific pass from Town captain Amanda Crump.
Town battled hard to respond in the second-half with Cossey firing just over the crossbar after drifting in from the left before Richardson was presented with perhaps Town's best opportunity to equalise, but her dipping volley flew just wide of the post.
Town continued to press with Lindsey Cooper next presented with an opportunity but she headed Crump's corner wide.
Despite their endeavour, nothing was coming off for the Tractor Girls, with many attempted crosses and through passes not finding their way into the path of the Town forward players.
The game eventually petered out and Norwich held on for their first victory over Town in four attempts.
Town XI: Fagg (GK), Ball, Duchar-Clark, Cooper, Pannifer (Arnoup), Welton, Wakefield (Craddock), Crump (c), Cossey, Richardson, Thomas