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Nathan Redmond made a bright start to the match as he looked to create an early goal for the visitors, but his effort inside the opening 10 minutes went well wide.
Januzaj had a couple of chances to double the reigning Premier League champions' lead before the break, curling an effort from the edge of the box just wide, before seeing his second attempt blocked just before the half-time whistle.
Redmond tried to get the Canaries back on level terms at the start of the second half, showing his pace before delivering the ball for Johan Elmander, who flicked the finish marginally wide.
Hernandez punished the visitors for their miss three minutes later, bundling the ball past Mark Bunn for his second goal of the evening, having seen the Norwich keeper deny him moments earlier.
However, play was halted for around 10 minutes following the chance, as Snodgrass appeared to pick up a head injury following a collision with Rafael while attempting the header, resulting in the Norwich man leaving the pitch on a stretcher.
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David Moyes's side continued to looked strong as full time approached, and were able to add their third goal when Phil Jones volleyed in, while Fabio also got on the scoresheet deep into stoppage time just moments after coming off the bench, tapping in from close range.
United will learn their opponents for the last eight of the tournament when the draw takes place tomorrow evening.