Rail Minister Jo Johnson said the money is "short term repayable financing" as an "interim measure" which will ensure "full momentum is maintained" behind the late-running project.
The funding will go towards completing infrastructure in the tunnels, work on stations and testing of safety and reliability.
Yorkshire MPs heavily criticised the decision.
Bradford South MP Judith Cummins told The Yorkshire Post: “Transport infrastructure in the North is at breaking point but yet again, the Department for Transport has managed to find millions down the back of the sofa for London.
“Crossrail is late and over it’s already vast budget. It has had money repeatedly thrown at it - meanwhile we are given spare change for projects that often never materialise.
“The day when we have the proper funding and powers to truly deliver on transport for people in the North cannot come soon enough.”
Hull North MP Diana Johnson said the move was "grossly unfair" given the Â£15bn already spent on Crossrail, and given her city has had a bid for a slice of the Transforming Cities transport fund rejected.
The Labour MP said: “In terms of scale, Â£350m is enough to complete the Selby to Hull rail electrification scheme that was blocked by Ministers two years ago - even though it was privately financed - three times over.
"This just adds to the suspicion that, under this Government, other transport investment schemes in the South East, such as the Â£32bn Crossrail 2, will happen far ahead of any real improvements in the North.”
Former Transport Secretary Lord Adonis criticised Transport Secretary Chris Grayling.
"This is a serious further failure of management at Crossrail and is costing passengers and taxpayers dear in delays and huge extra costs," he said.
"It is time Chris Grayling got a grip."
Luke Raikes, senior research fellow at the IPPR North think-tank said: “Yet again the Department for Transport backs London with a real financial commitment, while the North is left with little cash.
“Under Government plans, London will receive five times more transport investment per person than Yorkshire and the Humber. And despite Government promises, they have yet to move forward with electrifying the line connecting Manchester and Leeds – two of the UK’s most important cities. Every day, people in the North suffer the effects of these decisions with no end in sight.
“No one would deny necessary investment to London. But the North is still waiting for a clear and significant commitment to the Northern Powerhouse we’ve heard so much about for almost five years."
It was announced on August 31 that the capital's new east-west railway will open in autumn 2019 rather than December this year to complete infrastructure and testing.
The project's budget was increased by Â£600m in July due to "cost pressures". It is being predominantly funded by Transport for London (TfL) and the Government.
Mr Johnson said London will bear "any additional costs via a financing arrangement" because the city is the "primary beneficiary" of the scheme.
He also announced that TfL and the Department for Transport have commissioned an independent review of Crossrail's governance and a separate review of its finance and commercial position.
Trains were due to operate through the central tunnels from Paddington to Abbey Wood from December, when separate services on the Paddington-Heathrow and Liverpool Street-Shenfield routes would continue.
In May 2019, direct trains from Paddington to Shenfield were due to launch, with the line fully open from Reading and Heathrow in the west to Shenfield and Abbey Wood in the east from December 2019.
Once the service begins it will be known as the Elizabeth line.
Speaking to reporters in Brussels on Friday morning, London Mayor Sadiq Khan said: "I was angered, disappointed and frustrated when I was told by Crossrail Ltd that the central section of Crossrail wouldn't open in December as had been promised for a number of years now.
"It's quite clear that as a consequence of that delay there are additional costs to be expended.
"I have met with the Chancellor, I have spoken on a couple of occasions with the Secretary of State for the DfT.
"As joint sponsors we agreed this amazing project has got to be finished."
London's transport commissioner Mike Brown said: "The confirmation of an interim financing package between the Government and the Mayor of London will enable Crossrail Ltd to continue its construction work and vital testing at pace to open the Elizabeth line to passengers as quickly as possible."
Caroline Pidgeon, chairwoman of the London Assembly Transport Committee, said: "We welcome this announcement by the Government, but it is only a sticking plaster to keep the project going.
"Londoners need to know what work is left to open the line, how much this will cost and ultimately who will pick up the tab. This is a joint project with Government and all partners need to pay their fair share."