A 75th minute header from Ciaran Clark and a last minute strike from Idrissa Gueye settled the tie as Villa, beaten finalists last season but currently bottom of the Premier League, won for only the fourth time this season to set up a home meeting with Manchester City.
West Bromwich Albion, the only other Premier League team in action, were also made to battle for victory over lower league opposition, seeing off Championship (second tier) strugglers Bristol City 1-0 at Ashton Gate.
The home side, who sacked manager Steve Cotterill last week, kept the Baggies out until the 52nd minute when Venezuela international Salomon Rondon, West Brom's record 12.0 million pounds ($17.00 million) signing, trapped the ball on his chest before sweeping it home.
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West Brom will face third tier Peterborough United at home in the next round.
Like Villa, debt-ridden Bolton Wanderers are another club whose glory days seem long gone but they also enjoyed a much-needed win.
Four-times FA Cup winners Bolton, struggling at the bottom of the Championship and facing a fight for survival with debts of 172.9 million pounds, won their replay against minor league Eastleigh 3-2 at the Macron Stadium.
Eastleigh, who play in the fifth tier National League, gave Bolton an early scare when they took the lead after 11 minutes through fullback Joe Partington.
Bolton, who face a winding-up order from the tax authorities at the High Court next month, came back by scoring twice in four minutes through Gary Madine and Dean Moxey just before halftime.
The visitors equalised through Kaid Mohamed before the break before Darren Pratley, who scored the late equaliser in the original tie, scored what proved to be the winner after 52 minutes. Bolton now face Leeds United at home in the next round.
There was also joy for Portsmouth, now playing in the fourth tier but FA Cup winners as recently as 2008, who saw off second tier Ipswich Town 2-1 at Fratton Park. They now face south coast rivals Bournemouth.
(Reporting by Mike Collett; editing by Justin Palmer)
