It was cruel on the hosts, who were the better side right until it came down to spot-kicks.
Keeper Brad Guzan was the hero for the visitors saving three penalties during the shoot-out and one during normal time.
Sunderland could and should have won it in normal time. They should have won it in extra-time, but missed opportunities were to come back to haunt them.
Lorik Cana, Jordan Henderson and Andy Reid all saw their efforts saved by Guzan, who got everything right on the night.
But in truth had the Steve Bruce's men took their opportunities the game would never have gone to penalties.
With the clock running down in the second half of extra-time Kieran Richardson had the chance to put his side through to the last 16.
The midfielder latched on to a long clearance and bore down on goal, but his shot was straight at Guzan.
Maybe the Black Cats should have realised their luck was out when they passed up another gilt-edged chance in normal time.
Referee Phil Dowd pointed to the penalty spot on 83 minutes after a handball by Carlos Cuellar.
With regular penalty taker Darren Bent rested, up stepped Kenwyne Jones, but his penalty was saved as Guzan guessed right and dove to his left to make a good stop.
The Black Cats could also have scored in normal time just seconds before the penalty, as James Collins almost put through his own net - only for Guzan to come to the rescue once more.
A goal would have been just reward for the Wearsiders, who while not outstanding were the better of the two teams.
Villa seemed content to wait and try to hit their hosts on the break, but those breaks were far too sporadic for much of the encounter.
Sunderland on the other hand were a threat for most of the game and asked questions of Guzan time after time, only for the American to pass the tests with flying colours.
Henderson's shoot on sight policy also paid dividends with the teenager going close on several occasions.
The midfielder scored the opener in the previous round against Birmingham City, and he looked the man most likely in the first half against Villa.
His first chance came after just 12 minutes. Midfield partner Lorik Cana teed Henderson up on the edge of the area, but the youngster fired straight at Guzan.
Just before the half hour mark Henderson tried his luck again, this time firing just wide from about 20 yards out.
But his best effort came right on the stroke of half-time. Cana crossed for Jones, but Richard Dunne stooped to head clear.
The ball only reached Henderson on the edge of the 18-yard box and his powerful half-volley looked destined to find the target until Guzan pulled off an excellent save.
Henderson continued in the same vein after the restart, and went close again on 56 minutes, his chip dropping just wide after good work by Jones played him in.
Villa were brighter in the second half, but they never really tested Craig Gordon in the home goal.
In the end though that mattered little as the visitors booked their place in the last 16 as some criminal finishing by Sunderland got them out of jail.