Blackstock ended a run of four defeats, three of them under new boss Burley, by scoring twice to finish off weakened Brighton.
Southampton went into the match without ten injured players, but they were still better off than Brighton who were deprived of the services of 12 players.
In addition the manager Mark McGhee chose to leave out little striker Leon Knight from his starting line-up.
Blackstock struck for the first time in the tenth minute when Martin Cranie burst through from the back to set up a shooting chance which he finished off with a left-foot shot just inside a post.
But plucky Brighton made light of their injury problems by equalising after 21 minutes.
Alex Frutos and Jake Robinson combined on the left and Kerry Mayo scored his first goal of the season with a low right-foot shot.
Djamel Belmadi hit the top of the Brighton bar after 38 minutes and later Mark McCammon cleared from the Brighton line a goalbound shot from Blackstock.
But the turning point of the match came in the 61st minute when Antti Niemi dived to his right to make a tremendous save from a long-distance shot by Frutos.
Florent Chaigneau made an equally important stop to push a dipping shot from Belmadi over the bar and then got down to keep out a low effort from Chris Baird.
But Southampton's pressure paid off with four minutes remaining when Nathan Dyer at last got the better of 17-year old Joel Lynch to centre for Blackstock to score from ten yards.
Lynch was making his debut and there was a chance in the last minute for Joe Gatting to also get his first taste of Championship football, but this time there was no way back for Brighton.