The BBC's media editor Amol Rajan said Mr Osborne, who is to edit the paper four days a week, intended to stay on as MP for Tatton, in Cheshire.
But Mr Osborne is facing calls to quit politics altogether, with Jeremy Corbyn calling the appointment a "joke".
Labour MP John Mann said Mr Osborne - who was sacked as chancellor by Theresa May in July - was "pretending" to be an MP and becoming a full-time newspaper editor on top of being an MP "devalued Parliament".
The Standard's proprietor, Evgeny Lebedev, said Mr Osborne was "London through and through" and he was confident that the MP was "the right person to build on the fantastic legacy of Sarah Sands".
"I am proud to have an editor of such substance, who reinforces The Standard's standing and influence in London and whose political viewpoint - socially liberal and economically pragmatic - closely matches that of many of our readers," tweeted Mr Lebedev.
'Surprise'
The newspaper said its schedule would enable Mr Osborne to "continue to fulfil his other commitments, including as an MP; giving him the time to vote and contribute in Parliament in the afternoon after the paper has gone to print, and be in his constituency".
Mr Osborne, who earlier addressed Standard journalists in the paper's news room, said it was a "great honour" to be asked to take charge of one of the UK's most influential newspapers.
He sought to play down concerns about his political neutrality, saying he would be "fearless" in speaking up for Londoner's interests.
"Growing up as a Londoner, I've always known that the Evening Standard is an institution that plays a huge part in the life of the city and its people," he said.
"I am proud to be a Conservative MP, but as editor and leader of a team of dedicated and independent journalists, our only interest will be to give a voice to all Londoners.
"We will judge what the government, London's politicians and the political parties do against this simple test: is it good for our readers and good for London? If it is, we'll support them; if it isn't we'll be quick to say so."
Rajan, a former editor of The Independent, which is also owned by Mr Lebedev, said the appointment was a "fantastic coup" for the newspaper but would "take a considerable amount of people by surprise".
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