Max, the character, used the disgraced presenter's accent and catchphrases and wore one of Savile's trademark tracksuits topped off with a blond wig.
The episode featured Max presenting a Top Of The Pops-style programme and uttering the paedophile's catchphrase: "Now then, guys and gals."
The episode, which was filmed in 2001, was shown on the BBC before 9am today.
The Tweenies were co-created and produced by Scottish actor Iain Lauchlan.
A corporation spokesman said: "This morning CBeebies broadcast a repeat of an episode of the Tweenies, originally made in 2001, featuring a character dressed as a DJ impersonating Jimmy Savile.
"This programme will not be repeated and we are very sorry for any offence caused."
Twitter users picked up on the gaffe.
Kenny Senior wrote: "Are BBC trying to self destruct? Max from Tweenies dressed as Jimmy Savile just now nearly chokes on my cornflakes."
A recent Metropolitan Police report into allegations of sexual abuse against Savile, who died in 2011 aged 84, concluded he was a "prolific, predatory sex offender" who abused more than 200 people over a 60-year period.