Two goals from Jermain Defoe and an Emmanuel Adebayor penalty was enough to confirm Spurs' place in the next round, after the League One promotion hopefuls took an early lead with just four minutes on the clock.
The north Londoners now have a home clash against Premier League opponents Bolton Wanderers to look forward to as the club continue their hunt for cup glory.
"It was a difficult game, just what I expected," Redknapp told Tottenham's official website. "When I heard people saying we would smash this lot at home I thought 'not a chance'.
"You only had to look at their record in terms of goals conceded away from home, very low, best in the division and a well-organised team who work their socks off.
"They are also dangerous on set-plays, every free-kick and corner comes in with a great delivery. They've also got some real competitors and they got off to a really good start with the penalty.
"I thought then 'this is going to be a hard night' but we came through in the end. Full credit to the lads, they showed good character to come back from 1-0 down. In the end we played some good stuff and got through."
Tottenham's performance was marred by a nasty looking injury to defender Michael Dawson, who was stretchered off and rushed to hospital with suspected medial ligament damage.
Aaron Lennon also limped off in the first-half with a hamstring injury, but it is Dawson who is causing Redknapp more cause for concern, having only recently returned from a four-month spell on the sidelines.
"We're not sure yet (on the extent of the injury) but it didn't look good," he concluded.
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