
City and United, currently first and second in the Premier League, took contrasting routes to the last 16, where they were joined by Chelsea, Arsenal, Tottenham Hotspur and Everton as the big guns advanced despite fielding weakened teams.
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United manager Jose Mourinho made nine changes from the side that thrashed Everton on Sunday, with Rashford and Juan Mata the only survivors.
Rashford took just five minutes to open the scoring with a deft finish and the England forward struck again in the 17th minute with a powerful effort.
Burton, 19th in the Championship, were killed off in the 36th minute when Jesse Lingard's shot deflected in off Ben Turner.
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Injury-ravaged United left-back Luke Shaw came on for his first appearance since April before Anthony Martial scored the fourth from Rashford's pass on the hour.
At the Hawthorns, City boss Pep Guardiola made eight changes from the 6-0 demolition of Watford, giving Yaya Toure his first start of the season, while Ilkay Gundogan, back from a serious knee injury, started for the first time since December.
It was from Gundogan's saved shot that Leroy Sane slotted in City's third minute opener.
But, casting a shadow over City's progress, Gundogan was forced off with a knock in the second half.
Claudio Yacob equalised for Albion with a close-range finish in the 72nd minute, only for Germany winger Sane to produce a superb curling winner five minutes later.
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