Villa will play Bolton Wanderers for a place in the final on February 29 in Cardiff, while Middlesbrough, who beat Tottenham Hotspur 5-4 on penalties in Wednesday's other quarter-final, will play Arsenal.
A fantastic goal by Colombian striker Juan Pablo Angel gave Villa a 16th-minute lead, Chelsea's England playmaker Joe Cole levelled in the 69th minute but Gavin McCann lashed home the winner nine minutes later after an Angel shot had been blocked.
The result was a huge upset, with Villa lying just a point above the relegation zone and third-placed Chelsea challenging for Premier League and Champions League honours.
Chelsea's squad has been assembled at a cost of 111 million pounds ($194.7 million) since the July takeover by Russian billionaire Roman Abramovich and between the starting XI and the substitutes, coach Claudio Ranieri fielded virtually all his best players at Villa Park.
More worrying for Chelsea is that Wednesday's setback followed Saturday's first home league defeat of the season, a 2-1 loss to unfancied Bolton.
"In this period, everything is going wrong for us," Ranieri told Sky Sports television. "We tried to win, but Villa played better than us and deserved to win, honestly."
"When we drew level I was sure we would win. But Villa had a very big heart."
Villa boss David O'Leary said: "They had a bit of belief in themselves and did what you have to do against an excellent side like Chelsea, which is to work very hard."
Chelsea's problems started when defender William Gallas mis-judged a header over 30 metres out. Angel seized possession, dribbled past John Terry and smacked a swerving shot past keeper Neil Sullivan. Substitute Hernan Crespo teed up Chelsea's equaliser, knocking the ball back for the inrushing Cole to steer past keeper Thomas Sorensen.
But McCann rifled in the winner and Chelsea were twice denied in the dying minutes when Sorensen saved a Terry header and Cole volleyed into the side-netting.
Darren Anderton put Spurs in front after just 62 seconds when Boro failed to clear a cross, only for Michael Ricketts to level four minutes from time.
Deadlocked after extra time and level at 4-4 in the shootout, French defender Franck Queudrue converted Boro's winning penalty after Spurs' Argentine defender Mauricio Taricco hit the post. Boro also beat Everton 5-4 on penalties in the previous round.
Boro boss Steve McClaren said: "Practising their penalties, which they do every day, has paid dividends against Everton and now against Spurs."