United, who have not won a trophy since 2017 and were playing their first game since Cristiano Ronaldo’s departure, proved too good for the Championship leaders at Old Trafford.
Eriksen netted the opener after 27 minutes. Portugal star Bruno Fernandes was the architect, his fine diagonal ball hooked back across goal by Aaron Wan-Bissaka for Denmark’s Eriksen to smash in first time from six yards.
Rashford, who hit three goals for England in Qatar, added a superb solo effort on 57 minutes, driving hard at the Clarets defence before firing low into the far corner.
Brighton were dumped out in a penalty shoot-out by League One strugglers Charlton who, with new manager Dean Holden watching from the stands, won the spot-kick duel 4-3 after the game ended 0-0.
Wales international Brennan Johnson netted twice, while Jesse Lingard and Taiwo Awoniyi also scored, as Nottingham Forest eased to a 4-1 victory at Championship promotion contenders Blackburn.
Rovers, who opted to select a second-string side, had levelled through Scott Wharton.
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