Cameron Jerome had Norwich's best chance with a header only to be denied by the excellent David de Gea but a mistake from Sebastien Bassong allowed Wayne Rooney to burst through and find Mata, who provided an excellent left-foot finish.
In a game of few chances and little excitement, United were left content with a victory that moves them to within one point of fourth-placed Manchester City.
Norwich, though, are plunged into deeper turmoil in 19th place with just two games left.
(Reporting by Ian Chadband; Editing by Toby Davis)
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