In earlier action Bolton Wanderers and Tottenham Hotspur battled to a 1-1 stalemate at the Reebok Stadium after Tom Huddlestone missed a second-half penalty for the visitors.
Fulham though had no trouble easing their way into the last eight for the second successive season, the Premier League side scoring a 4-0 win over Notts County of League Two, the lowest ranked club left in the competition.
County's miserable week, in which they were sold and saw former England coach Sven-Goran Eriksson leave as Director of Football, ended in almost inevitable disappointment as Fulham won with goals from Simon Davies, Bobby Zamora, Damien Duff and substitute Stefano Okaka at Craven Cottage.
Villa appeared set to become the latest big club after Manchester United, Arsenal and Liverpool to be eliminated when a stunning 30-metre free-kick from Darren Ambrose put Palace 2-1 ahead after 70 minutes.
The south London club, in administration and up for sale, were indebted to goalkeeper Julian Speroni for keeping them in the competition with a series of great saves but he could not stop Petrov's glancing 87th-minute header which saved the League Cup finalists from elimination.
Johannes Ertl had earlier headed Palace ahead after 24 minutes before Danny Collins scored Villa's first equaliser 11 minutes later.
PASSING MOVE
Bolton deservedly took the lead against Spurs at the Reebok Stadium when Kevin Davies completed a flowing 16-man passing move with a well-taken strike after 34 minutes, choosing his spot perfectly as goalkeeper Huerelho Gomes scrambled back to cover.
The home side dominated the first half but Spurs took control after the break and hit the bar twice in two minutes through a Peter Crouch header and a Paul Robinson back-pass before Defoe smashed home their equaliser after 61 minutes.
It was his 21st goal of the season and fifth in four FA Cup ties.
Huddlestone then blew his chance to win the match in the 71st minute when Bolton keeper Jussi Jaaskalainen saved his weak penalty, continuing Spurs woeful record with spot-kicks this season.
Manager Harry Redknapp stripped Defoe of the responsibility after scoring only four of his last 10 penalties for Spurs and his previous club Portsmouth.
Spurs assistant coach Joe Jordan said afterwards: "We drew 2-2 with Everton and missed a penalty, missed one in the Cup against Leeds and there have been other occasions. So, we need a bit more practice I think."
In Saturday's FA Cup action there were wins for holders Chelsea who beat Cardiff City 4-1, while Premier League Portsmouth and Birmingham City also made the last eight.
Manchester City and Stoke City drew 1-1 while Reading and West Bromwich Albion drew 2-2 in an all-Championship tie.
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