The Belgium international, who was selected for the youth side along with Memphis Depay and Phil Jones, was forced off along with Guillermo Varela in the first half on Monday
Januzaj was picked by Warren Joyce for the meeting with the Canaries along with the likes of Memphis Depay, Phil Jones and Sergio Romero.
But the former Borussia Dortmund loanee failed to complete the first half and was replaced by Matthew Willock after 33 minutes after failing to run off the problem.
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In a further blow to the home side, defender Guillermo Varela sustained a similar injury and was replaced by Timothy Fosu-Mensah, with both players heading straight down the tunnel instead of taking their places on the bench.
The injuries come at a particularly troublesome time for United, with Luke Shaw, Antonio Valencia, Marcos Rojo, Ashley Young and Bastian Schweinsteiger all currently sidelined.