The travel business said it has "entered into binding agreements" to purchase the operations, which also include a deport at Lea Interchange, for an initial £10 million followed by £1 million each year for 10 years after the move is completed.
Kelsian's east London operation operates 11 contracts on behalf of Transport for London (TfL), using a fleet of around 150 buses, with an annual turnover of around £38 million.
It said the move is "a good strategic fit" alongside Stagecoach's existing London operations.
Stagecoach added that the new depot will also provide the business with a platform to bid for new contracts in the area.
The group said it expects its net debt to increase by around £20 million upon completion of the move.
Martin Griffiths, Stagecoach chief executive, said: "We have an extensive track record of operating high-quality and profitable bus services on behalf of Transport for London.
"This targeted acquisition provides operational and margin improvement opportunities that we have successfully delivered at our other London bus depots.
"We look forward to working with TfL to deliver even better bus links to local communities in east London and also helping support the Mayor's drive to deliver a net zero transport system."
It comes as Stagecoach looks set for a £595 million takeover by German investment fund DWS.
The travel company's board backed the move in favour of a rival merger deal from coach operator National Express.
Last week, National Express confirmed it will not raise its bid for Stagecoach and maintained it believed the move is "superior" to DWS's offer.
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