England Under-21 international Clyne agreed a four-year deal last Thursday but the clubs continued to negotiate a compensation fee, reported to be £2.5million.
The 21-year-old full-back played for Saints in last weekend's 1-0 victory over Evian and could participate against Ajax on Saturday.
"Nathaniel has done very well," said manager Nigel Adkins.
"He only had one training session before he joined us so we're upping the game with him in a way that will make sure he doesn't break down."
Clyne's contract with Palace had expired but Saints were obliged to pay a compensation fee due to his age.
The Selhurst Park academy product made 139 appearances for Palace.
"We are also very pleased to have worked constructively with Southampton to conclude the deal before it went to a tribunal," said Palace co-chairman Steve Parish.