The victory goal - 2-1 and 3-2 total over two semi-finals against Leicester - came in the 93rd minute.
Villa's supporters, who did not have much to celebrate this season, became so much that they stormed the pitch at Villa Park in the thousands.
Matt Targett gave Villa the lead already in the 12th minute, but when Kelechi Iheanacho acknowledged a good shot in the 72nd minute, the meeting was perfectly even and seemed to be settled on penalty kicks.
In the final, Aston Villa will face Manchester City or Manchester United. City are 3-1 up for Wednesday night's return at home.