The Championship side will visit either Manchester City or Aston Villa in the last eight, with the Premier League duo facing one another on Wednesday evening for the right to host Brian McDermott's side.
Mills struck the only goal in the 26th minute, following up after his initial header from a corner had been blocked.
Lacklustre Everton, beaten by Chelsea in the 2009 final, created few clear-cut openings.
Their best chance fell to Leon Osman late on but he was denied by a fine block from Reading goalkeeper Alex McCarthy.
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