A record crowd of 7,999 packed into Withdean in the hope of seeing Brighton repeat their 2-0 victory at Millwall last Saturday.
Manager Mark McGhee went again with an attack minded 4-3-3, but it was Southampton who totally dominated the first half.
The visitors created a string of chances, beginning with a shot by Chaplow straight at keeper Wayne Henderson after a mix up in the Brighton defence.
Grzegorz Rasiak headed straight at Henderson from a corner and Djamel Belmadi missed the target when the Brighton keeper failed to hold a shot from Fuller.
The breakthrough finally arrived in the 37th minute as Fuller slotted his third goal in two games from 10 yards when the Brighton defence failed to clear.
The home side improved after the break as Southampton's veteran keeper Kevin Miller was finally forced into serious action.
Miller saved a Joel Lynch header but he should have been beaten on the hour when Colin Kazim-Richards was clean through but the young striker delayed and was crowded out.
That miss proved costly as Chaplow wrapped up Southampton's first away win since October three minutes later. The midfielder side-footed in a low cross from Belmadi.
Brighton to their credit refused to give in. Kazim-Richards tried to atone for his miss with an amazing run through several challenges which set up Gifton Noel-Williams for a shot that Miller saved with his right hand.
Brighton's French substitute Sebastien Carole spurned a late chance to pull a goal back, volleying over a Gary Hart cross when completely unmarked.
Manager Mark McGhee went again with an attack minded 4-3-3, but it was Southampton who totally dominated the first half.
The visitors created a string of chances, beginning with a shot by Chaplow straight at keeper Wayne Henderson after a mix up in the Brighton defence.
Grzegorz Rasiak headed straight at Henderson from a corner and Djamel Belmadi missed the target when the Brighton keeper failed to hold a shot from Fuller.
The breakthrough finally arrived in the 37th minute as Fuller slotted his third goal in two games from 10 yards when the Brighton defence failed to clear.
The home side improved after the break as Southampton's veteran keeper Kevin Miller was finally forced into serious action.
Miller saved a Joel Lynch header but he should have been beaten on the hour when Colin Kazim-Richards was clean through but the young striker delayed and was crowded out.
That miss proved costly as Chaplow wrapped up Southampton's first away win since October three minutes later. The midfielder side-footed in a low cross from Belmadi.
Brighton to their credit refused to give in. Kazim-Richards tried to atone for his miss with an amazing run through several challenges which set up Gifton Noel-Williams for a shot that Miller saved with his right hand.
Brighton's French substitute Sebastien Carole spurned a late chance to pull a goal back, volleying over a Gary Hart cross when completely unmarked.