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Manchester City great Kompany enjoyed numerous derby victories over United but his Red Devils reunion as Burnley manager ended in a fourth-round defeat on Thursday (AEDT).
Eriksen opened the scoring with a 27th-minute strike before Rashford doubled the advantage in the second half of United's first game since the World Cup.
Championship leader Burnley impressed in parts but was punished for wasteful finishing as Erik Ten Hag's side booked their place in Friday's (AEDT) last-eight draw.
Alejandro Garnacho squandered a glorious eighth-minute chance as Bailey Peacock-Farrell raced out to keep out a tame finish after Bruno Fernandes' incisive pass.
Fernandes provided another defence-splitting diagonal ball for Aaron Wan-Bissaka, who acrobatically hooked across for Eriksen to tap home a deserved opener.
A curled Manuel Benson strike forced a smart save from Martin Dubravka, who almost turned into his own goal from the resulting corner, before Peacock-Farrell pushed away to deny Antony Martial at the other end.
Scott McTominay whipped a presentable opening over after the interval, but Rashford extended United's lead when his powerful run ended with a drilled finish into the bottom-left corner.
Darko Churlinov wastefully prodded wide on the counter-attack and Ashley Barnes blazed over as a profligate Burnley showing ensured there was no way back for the Clarets.
