More than 40,000 viewers cast their votes by post, phone or email and chose the cuddly pink and yellow striped cat, ahead of shows like The Teletubbies, The Magic Roundabout and Play School.
Bagpuss was born in 1974 and was originally shown as part of the Watch With Mother series. Only 13 episodes were made.
Created by Smallfilms, which also made Noggin The Nog, Bagpuss told stories of a magical cloth cat who slept in a shop window.
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Each episode, Emily, the young girl who lived at the shop give the cat an object that she had found.
She would wake up Bagpuss who would then wake his friends - including Professor Yaffle, Gabriel the Toad and the mice from the Mouse Organ - who would identify and repair the item.
Every programme ended with the words: "Bagpuss gave a big yawn, and settled down to sleep. And when Bagpuss goes to sleep, all his friends go to sleep too. The mice are ornaments on the Mouse Organ, Gabriel and Madeleine are just
dolls. And Professor Yaffle is just a carved wooden book-end in the shape of a woodpecker. Even Bagpuss himself, once he is asleep, is just an old, saggy cloth
cat - baggy and a bit loose at the seams. But Emily loved him."
'Place in our hearts'
Lorraine Heggessey, Head of BBC Children's Production, explained why the big cuddly cat is still a hit today. "It's always programmes we watched when we were youngest that have a special place in our hearts," she said.
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"Bagpuss is a favourite with today's twenty-somethings and it's great to see that they are just as keen to wallow in nostalgia as the rest of us. From Bill and Ben to The Teletubbies, the BBC continues its tradition of delighting younger viewers."
Bagpuss was declared the winner on BBC1 at 16.05 GMT on Friday during Are You Sitting Comfortably? - a tribute to children's programmes past and present.
The stripy cat beat off tough competition to land the prize including:
Rhubarb and Custard a wobbly cartoon of a cat and dog embroiled in a battle of wits.
Andy Pandy was one of the original Watch With Mother shows. Only 26 episodes were made of the original series, shown between 1953 and 1958, although they were repeated up until 1968. The success of the repeats led to a further 13 being made in colour in 1970.
Trumpton was always remembered for Captain Flack's rollcall of firefighters:
"Pugh, Pugh, Barney McGrew, Cuthbert, Dibble, Grubb". The fictional town was the capital of Trumptonshire, with nearby villages Camberwick Green and Chigley having their own separate series.
The Magic Roundabout was originally created for pre-school French children. The show's script was completely re-written by Emma Thompson's dad Eric.
Mr Benn lived at 52 Festive Road. He dressed up in a variety of costumes and went off on different adventures.
The Herbs featured Sir Basil, Lady Rosemary, Dill the dog who chased his own tail, Bayleaf the gardener and Parsley the lion, who ended up with his own spin-off series.
The Clangers were pink woolly creatures who lived on a small blue planet. They survived on blue string pudding and bowls of soup from the Soup Dragon and spoke in a language of musical whistles.