Redknapp made nine changes for the second-round tie and paid the price as Adam McGurk secured a 1-0 win for the underdogs with a superb 77th-minute free-kick at the Pirelli Stadium.
The result increased the gloom surrounding promoted Rangers' start to the season after defeats in their opening two Premier League games.
Redknapp said: "It was a poor performance
We gave the lads who haven't been playing an opportunity to show what they can do, and they showed me what they can do - and it wasn't an awful lot.
"We had chances in the first half and I thought we could have gone on to win the game but we faded in the second half and I was disappointed."
Rio Ferdinand, Steven Caulker, Joey Barton and Loic Remy were among the high-profile names given a night off while goalkeeper Robert Green sat on the bench.
Asked if he had any regrets, Redknapp said: "Not at all
We have got a massive game on Saturday (against Sunderland).
"Every team in this cup competition has made changes
Aston Villa have been beaten at home by Leyton Orient, Shrewsbury went to Leicester and won, MK Dons beat Man Utd.
"It is the way the competition has gone
You give lads a chance to play but obviously there are an awful lot of lads at Premier League clubs who are really very lucky to be there."
The only bright spot Redknapp identified was the first-half performance of playmaker Adel Taarabt, who after loans at Fulham and AC Milan was making his first appearance for the club in 16 months.
With the transfer deadline approaching, uncertainty remains over the 25-year-old Moroccan's future.
Redknapp said: "I thought he was different class in the first half
He got a bit tired towards the end but I thought his performance was encouraging
If we can get him fit he is a real asset.
"I don't know what he will do
He has got a buy-out clause but we saw in the first half that the ability he has got is on another level
When he wants to play he is a fantastic talent."
While it was a forgettable night for Rangers, whose best effort was a Junior Hoilett shot against the bar, it was a famous one for relative League newcomers Albion.
The Brewers are unbeaten this season, two points off the top of League Two after four games, and overcame Wigan in the previous round.
Manager Gary Rowett said: "It is absolutely amazing
The players were brilliant in everything they did
We limited them to a few long-range shots and not that many clear-cut chances.
"I thought we played with some really good attacking flair when we broke and counter-attacked, with some real pace and purpose
At times we opened them up quite well.
"We defended amazingly and kept our shape really well
We intercepted things and won balls
Sometimes, cyncially, we made fouls
That doesn't look good but it is part of the game.
"There were some brilliant performances in there and a moment of brilliance from Adam at the end.
"I was really pleased and I think the players, and everyone associated with the club, should be proud of what they have achieved."
Source : PA
Source: PA