Vine called the show - which features viewers' letters and complaints - "a prime slice of BBC real estate".
"I'll make sure we continue to hold the people at the top of the Beeb to account in the same way I've always done with politicians on Newsnight."
Points Of View was first aired in 1961 with Robert Robinson as its presenter. His successors include Barry Took, Alan Titchmarsh and Weakest Link host Anne Robinson.
"For as long as I can remember I've watched Points Of View doing the sort of thing other programmes have only just stumbled on - letting the viewer take the reins," said Vine.
The show, which returns on 20 April, will be filmed at the BBC's White City offices after years of being based in Birmingham.
Nick Patten, head of studio at BBC Birmingham, has thanked Sir Terry for his "marvellous" work. "He's been a stalwart for the series and an enormous pleasure to work with," he added.
Copyright Speakers Corner 2014