Following a goalless first half the game changed dramatically 15 seconds into the second period when Enckelman was sent off as he tried to block the progress of Brett Ormerod only to concede the spot kick.
Referee Steve Bennett was close at hand but it appeared that Alpay had impeded the former Blackpool striker.
Enckelman was replaced by Stefan Postma and the first thing the Dutchman had to do on his Premiership debut was to retrieve the ball from the back of the net after Beattie's successful penalty.
It was another bad night for Villa manager, Graham Taylor, who saw his team promise great things only to be torpedoed by the controversial penalty.
Villa's disgruntled fans were far from happy and several times shouted "Taylor out."
Southampton, to their credit, took full advantage of the situation to chalk up their first away win of the season.
Alpay, making his first appearance in five games, created an early impression with his powerful tackling to keep out the threat of Beattie and Ormerod but he blotted his copy-book in the penalty incident.
It was only a brilliant save from Enckelman that prevented Svensson opening his account.
Enckelman was again well positioned to save another individual effort from Ormerod and played with a great more confidence than of late.
Villa were often over anxious in front of goal with both Darius Vassell and Gareth Barry firing wide from good positions in a hectic first half.
Barry had the best chance when he was set up by Lee Hendrie only to see his left-foot angled shot brilliantly flicked away for a corner by Antti Niemi.
Niemi also got down well to save from Mustapha Hadji late on.
Villa have managed to score just six goals this season and this was the seventh game in which they have failed to score - statistics which led to the Villa Park crowd booing the team off the pitch both at half time as well as at the end of the game.