Nine Madryn Street, Toxteth, where the drummer lived until he was four, is among 271 homes which are due to be cleared under regeneration plans.
Housing Minister Grant Shapps had asked for alternative plans to be considered to renovate the homes.
The communities secretary still needs to approve the plans before demolition.
Eric Pickles is considering a request by campaigners for a full environmental assessment before the area is razed.
The street is one of several known as the Welsh Streets, which were built in the late 19th Century when there was an influx of Welsh people who came to Liverpool to work as dockers.
At a planning meeting at Liverpool Town Hall earlier, residents of the streets pleaded with the council to knock down what they called "damp slums" and replace them with modern homes.
Opponents of the demolition said the streets had historical importance to Beatles fans as well as potential tourist value.
The childhood homes of John Lennon - Mendips, in Menlove Avenue - and Sir Paul McCartney - in Forthlin Road - are preserved as tourist attractions supported by the National Trust.
George Harrison's Arnold Grove childhood home remains a private house.
A request for the Madryn Street house to be put under a preservation order was turned down by English Heritage last year, which ruled the property had no associations with the success of the Beatles as a group.
Liverpool City Council Leader Joe Anderson said the area was in dire need of regeneration.
He said: "A majority of the local community have expressed support for our plans since we started consulting on this plan over a decade ago.
"They are sick of living in poky, damp infested properties. They want and deserve decent, modern homes with gardens and space for a car and are exasperated and frustrated at the delays to this scheme."
He added: "I pledged last year that if a developer came forward with a viable proposal to retain the Welsh Streets then we would look seriously at their plans. I am sorry to say that this has not happened.
"It is simply not fair to keep the local community in limbo any longer.
"People have been voting with their feet for years and moving away from the Welsh Streets which is why the housing market in the area has completely collapsed."
Irene Milson, chair of the tenants' and residents' association, said of the decision to demolish the house: "I can't explain, I really can't explain, it's jubilation.
"It's been 10 years of hard work from the start to now, I just can't put it into words."
However Phillip Coppell, chairman of the Save Madryn Street campaign, said the fight was not yet over.
He said: "Demolition is not a foregone conclusion. We have people waiting in the wings who are prepared to redevelop the area and refurbish so we are still hopeful.
"It is an important part of the heritage trail. There are lots of plans to demolish the whole area but we believe that we should keep not only Ringo's house but the entire Welsh Streets."
Plans to demolish Beatle Ringo Starr's birthplace have been approved by Liverpool City Council, despite opposition from a government minister.