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The Argentine striker single-handedly destroyed Brighton's hopes of causing another upset following their second-round victory against Manchester City.
Villa scored at both ends against Nottingham Forest on Sunday after replacing Derby's injured top-scorer Rob Hulse.
With Hulse restricted to the substitute's bench by sore ribs, Villa took handsome advantage of his place in the starting line-up.
He opened the scoring in controversial circumstances in the 28th minute.
Brighton felt young goalkeeper John Sullivan was fouled by Villa when he dropped the ball in an aerial challenge.
Referee Kevin Friend saw nothing wrong with the challenge as Villa hooked the ball into the vacant net.
Nathan Ellington increased the lead six minutes later after out-muscling Brighton defender Colin Hawkins to rifle past Sullivan.
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The Seagulls pressed for an equaliser after the break but a substitution by Derby boss Paul Jewell proved pivotal.
He introduced Gary Teale - a recent loan target for Brighton - and the Scottish winger had a swift impact.
His cross within minutes of coming on was met by Villa at the far post with a stooping header.
Villa raced away from the Brighton defence to complete his hat-trick three minutes from time and book a home tie against Leeds United next week.