Redknapp could not conceal his delight as Liverpool never recovered from their early setback.
The visitors conceded a goal in the fifth minute when the ever-dangerous Peter Crouch put David Prutton clear and he beat goalkeeper Jerzy Dudek with a low right-foot shot.
Redknapp opted for a system of five midfield players leaving Crouch to battle on his own up front and it was a tactic that paid off handsomely.
Liverpool had still not mustered a shot on target by the time Southampton scored their second goal in the 22nd minute.
Jamie Redknapp put Prutton away on the right and his centre was headed in at the far post by Crouch.
There might have been another goal soon afterwards when Rory Delap raced clear but his shot was saved comfortably by the on-rushing goalkeeper.
At the start of the second half Liverpool were briefly revived, posing a threat which had been missing in the first half.
Milan Baros beat two men on the edge of the area and shot narrowly wide with goalkeeper Antti Niemi a bystander and the otherwise lethargic Steven Gerrard blazed a free-kick from the edge of the area against the top of the bar and over.
Fernando Morientes found it difficult to get into the match and it was Baros who posed the greatest threat to the Saints but Niemi was pressed into only one save, getting down to hold a first-half shot from Gerrard.
Redknapp had an outstanding match against his old club with some shrewd passing and important tackling but was booked for a first-half foul.
The Reds had Dietmar Hamann and Luis Garcia yellow carded for fouls as Liverpool struggled to get back into the match.
Florent Sinama Pongolle sent a header wide and the ever-dangerous Baros went close again for Liverpool with a shot which drifted wide but Southampton held on for a precious win.
Redknapp was justified in his tactical ploy although his hand was forced by injuries to leading scorer Kevin Phillips and to new signing Nigel Quashie who damaged knee ligaments in training on Friday.
MAN OF THE MATCH: David Prutton (Southampton) - Singled out by his manager for a whole-hearted performance, his runs from deep always posed problems to the Liverpool defence and his first league goal of the season was just reward for a fine display.