Chairman Ken Bates has already turned down two offers from unnamed Premier League clubs for Delph, a product of the Leeds Academy.
Delph demonstrated why he is so highly rated eight minutes from time.
Breaking from a Brighton corner, deep inside his own half, Delph ran virtually the length of the pitch before scoring with a left-foot curler.
Leeds should have gone ahead in the tenth minute when referee Pat Miller penalised Brighton midfielder Adam El-Abd for holding Argentinian striker Luciano Becchio from a corner.
Becchio took the penalty himself, but shot straight down the middle at Brighton goalkeeper John Sullivan, diving to his left, kept it out with his legs.
The home side, haunted by the threat of relegation, could have made Leeds pay for their generosity in the 25th minute.
Sullivan's long goal kick down the centre put Stuart Fleetwood clean through, but the striker, on-loan from Charlton, failed to beat Leeds keeper Casper Ankergren.
Chances were few and far between for both sides, but Leeds grabbed the lead in the 61st minute.
A free-kick was headed on by Becchio and Lee Trundle, making his full debut, on-loan from Bristol City, turned his marker to convert from close range.
Delph's stunning effort made sure of a fourth successive home defeat in League One for Brighton. Their best chance came in the closing stages when skipper Nicky Forster found the sidenetting after he was put through by substitute Dean Cox.
New Leeds boss Simon Grayson will be especially satisfied with a first clean sheet in 13 games after giving debuts to centre-half Richard Naylor, on loan from Ipswich, and left-back Carl Dickinson, who has been borrowed from Stoke.