It is hoped that the money - the bulk of which will come from the Greater London Authority (GLA) through a government borrowing facility - will cover additional costs to allow completion.
However, the timetable was on course to be put back further as no firm date could be committed to.
The Elizabeth Line - running from Reading and Heathrow in the west through central tunnels across to Shenfield and Abbey Wood in the east - was supposed to have opened this autumn.
Its independent report concluded the £15.4bn project could cost between £1.6bn-£2bn more, Transport for London (TfL) said.
It said that in addition to £1.4bn in a grant from the GLA, a further contingency loan facility of up to £750m was being made available by the Department for Transport.
TfL's statement said: "It has now become clear that more work is required than had been envisaged to complete the infrastructure and then commence the extensive testing necessary to ensure the railway opens safely and reliably.
"Today the new chief executive of Crossrail, Mark Wild, also confirmed that having reviewed the work still required to complete the project, an autumn 2019 opening date could no longer be committed to at this stage, and his team was working on a robust and deliverable schedule."
The GLA element of the funding will be repaid to the DfT through the levies on some business rates and on new property developments.
The extent of the problems mean it is now likely that no trains will run until 2020 at the earliest.
Crossrail, which will carry passengers from Paddington to Canary Wharf in 17 minutes, is one of Britain's biggest rail infrastructure projects for decades.
TfL has boasted that Crossrail will boost Britain's economy by an estimated £42bn while adding 10% to central London's rail capacity.