Villa have managed just 11 goals in 20 Premier League matches this season, scoring only one in their last five, and looked set for a replay at Bloomfield Road after struggling to break down one of the most porous defences in the Championship.
However, Benteke rifled home from the edge of the penalty area late on to give the home side an unconvincing win at Villa Park.
Paul Lambert's side upped their game after the interval, though, and Benteke's winner ended the hopes of the Championship's basement club.
The former Swansea City man again caused problems for the home defence moments later – goalkeeper Shay Given getting down well to keep out his deflected strike.
Villa had two chances to break the deadlock before the end of the half, but Hutton's effort deflected harmlessly into the arms of Joe Lewis, before Benteke hit the side-netting from a tight angle.
Former Villa man Nathan Delfouneso placed the ball wide from a promising position 10 minutes into the second half, only to see the linesman had already raised his flag for offside.
It took Villa until the 58th minute to get a shot on target – Benteke having a soft header saved by Lewis before Carlos Sanchez also tested the goalkeeper.
However, with home fans chanting 'We only want one goal', Benteke duly obliged, thrashing the ball in after cutting in from the left, somewhat banishing memories of last season's third-round embarrassment at the hands of Sheffield United in the same round.
