In-form striker Murray struck after 15 minutes at St Mary's and added a second five minutes later before Rickie Lambert pulled a goal back from the penalty spot just before half-time to give Alan Pardew's side hope.
But skipper Andrew Crofts added a third late on to consign Saints to their first defeat in nine games and gave Poyet a dream start to his managerial career.
Saints looked vulnerable in defence and the Seagulls capitalised after 15 minutes when Dean Cox crossed for Murray, who showed great technique to volley past Kelvin Davis with his right foot.
And the visitors stunned Southampton without a second goal five minutes later, when Nicky Forster set up the onrushing Murray and his initial shot was tipped onto the post by Davis, but the ball rebounded to Murray and the striker was left with a simple finish.
Saints were back in the game four minutes before half-time when Andrew Whing was adjudged to have fouled Adam Lallana in the penalty area and Lambert slotted home from the spot, but Crofts sealed the points five minutes from time when Liam Dickinson laid the ball into his path and he drilled a right-footed finish beyond Davis from the edge of the penalty area.
The win moved Brighton up one place to 19th, three points clear of the drop zone, while Saints - who began the campaign with a 10-point deduction - stay 22nd, four points from safety.