MTR was one of four companies short-listed to bid for the contract late last year.
The other train companies bidding to run the franchise included National Express, Arriva and French transport firm Keolis, through the Go-Ahead Group.
MTR is expected to employ 1,100 staff.
The franchise will run for an initial eight years with an option to extend the franchise for another 10 years after that.
London mayor, Boris Johnson, said the award of the franchise represented the "final pieces of the jigsaw puzzle" being put in place.
"I am delighted that we can now announce the new operator of this landmark railway, bringing it one step closer to welcoming passengers aboard," he said.
"Crossrail will revolutionise east-west transit in the capital, making London an even more attractive place to visit and invest."
The rail franchise will employ around 400 drivers and there will be more than 50 apprenticeships for people from communities along the route.
MTR already operates the London Overground franchise with Arriva UK Trains, as well as running the metro system in Hong Kong.
The huge Crossrail project, costing around £15bn to build, is the first complete new underground line in London since the Victoria line was built in the late 1960s.
The line will see trains running from Reading in Berkshire to the west of the capital and as far east as Shenfield in Essex, as well as linking to south London.
It is also one of the most ambitious transport construction projects undertaken in London since the original underground was built over 150 years ago.
Crossrail is expected to reduce passenger overcrowding on underground trains in central London and to reduce cross-London journey times.
TfL said it expected Crossrail to support the equivalent of 55,000 full-time jobs around the UK.
New trains will begin entering service in 2017. The route through Canary Wharf, the City and the West End will open in late 2018, with the full route running from late 2019.
MTR will start running services from the end of May 2015 between Liverpool Street and Shenfield, when it takes over the existing stopping services currently operated by Abellio Greater Anglia.
London Underground managing director Mike Brown said the award of the franchise was "a huge step towards the new railway coming to London."

Muslim Youth Focus Group organised the event to voice concerns about issues such as radicalisation.
The event was held as it emerged four teenage girls who were barred from travelling abroad were from the same school attended by three girls already thought to have fled to Syria.
All seven attend Bethnal Green Academy in Tower Hamlets.
The name of the school was revealed after reporters argued it was in the public interest.
The event was aimed at young British Muslims to help create a platform for them to discuss issues such as Islamophobia, Islamic State and the war in Syria.
In mid-February, police raised concerns following the disappearance of Kadiza Sultana, 16, Shamima Begum, 15, and Amira Abase, 15, from their homes in east London.
Shamima used the passport of her 17-year-old sister to leave the UK, police have said.
Young people, police, teachers, faith leaders and Muslims from the world of entertainment gathered at the Muslim Youth Festival to explore views on what it means to be a Muslim today.
Role models spoke at the event including music star Naughty Boy, whose real name is Shahid Khan, and has been working with Zayn Malik, who recently announced he was leaving One Direction.
Imam, Dr Mohammed Fahim of South Woodford Mosque said: "Young British Muslims, irrespective of their ethnicities or gender, should feel proud to live in a country like Britain which respects human rights and gives them equal opportunities and permits them to practice their religion without any fear or oppression."

She wore it to his 45th birthday celebration at Madison Square Garden in New York, but the garment was so tight she had to be sewn into it.
Monroe died less than three months later, and the president died in 1963.
The dress was sold in 1999 for $1.26m but Julien's Auctions thinks it could fetch up to $3m (Â£2.2m) this November.
The dress, made of silk gauze and covered in thousands of rhinestone beads and sequins, will go under the hammer in Los Angeles on 17 November.
Before then, it will be on show at Mana Contemporary in Jersey City, New Jersey, and the Museum of Style Icons in Newbridge, Ireland.
The £1.4bn contract to run the new Crossrail service through London from 2018 has been awarded to Hong Kong-based MTR Corporation, Transport for London (TfL) has announced.
[NEXT_CONCEPT]
Hundreds of young British Muslims have attended a festival in London on how to divert people away from extremism.
[NEXT_CONCEPT]
The flesh-coloured dress Marilyn Monroe wore when she breathlessly sang Happy Birthday to President John F Kennedy in 1962, is to be auctioned.