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Sir Alex Ferguson's side found it tough to break down a resolute Everton defence and looked in danger of dropping two points at Old Trafford, but Mexican striker Hernandez ended the stalemate with a close-range header seven minutes from full-time.
Hernandez's winner piles the pressure back onto title rivals Chelsea and Arsenal, who both trail United by nine points.
Second placed Chelsea host West Ham later on Saturday, while third placed Arsenal travel to Bolton on Sunday.
With a Champions League semifinal first leg clash against Schalke looming on Tuesday, Ferguson rested Nemanja Vidic, Patrice Evra, Ryan Giggs and Michael Carrick.
United still dominated the first half and Everton goalkeeper Tim Howard made two good saves to frustrate Hernandez.
Everton, who have not won at Old Trafford since 1992, felt aggrieved that referee Peter Walton didn't give a penalty early in the second half when Rio Ferdinand tugged back Victor Anichebe.
David Moyes' men almost snatched the lead when Jack Rodwell's long-range effort forced a sprawling save from Edwin Van der Sar.
United were struggling to breach Everton's defence. Even when they got through, Fabio da Silva's cross was turned onto a post by Toffees defender Sylvain Distin.
But they finally made the breakthrough in the 83rd minute when Antonio Valencia's cross to the far post was headed in by Hernandez from virtually on the goal-line.
