The company already works with TfL through London Overground operator LOROL, which it jointly owns with Arriva UK Trains.
It will employ around 1,100 staff to service the new contract, including almost 400 drivers, and provide apprenticeships for people living along the route.
MTR will start operating services from next May when it takes over services between Liverpool Street and Shenfield.
TfL says although these services will initially use existing trains, passengers will see immediate benefits including fully staffed stations and the ability of disabled passengers to turn up and go, with no need to pre-book assistance.
New trains, designed by Barber & Osgerby and built by Bombardier, will begin entering service from 2017.
The route through Canary Wharf, the City and the West End will open in late 2018, with the full Crossrail route running from late 2019.
Mike Brown MVO, Managing Director London Underground and London Rail, said: “Appointing an operator is a huge step towards the new railway coming to London. The decision to appoint MTR was reached after a thorough and competitive procurement process and we look forward to working with them to transform rail services in the capital.
“Crossrail will help cut journey times, make it easier for people travelling across the city and create jobs and support growth and regeneration along the route.”
Mayor of London, Boris Johnson, added: “I am delighted that we can now announce the new operator of this landmark railway, bringing it one step closer to welcoming passengers aboard.
“Crossrail will revolutionise east-west transit in the capital, making London an even more attractive place to visit and invest. As has been the trend throughout this project, many hundreds of additional jobs will be created for local people, with MTR expected to employ around 1,100 staff including almost 400 drivers.”
