He takes over from interim chief executive Keith Berryman and will start his job in May.
Haste, whose civil engineering experience includes running the design and construction of the Second Severn Crossing for Laing, joined the Terminal 5 team in 1996.
The scheme was given government backing in December.
Haste will be succeeded at BAA by former Stansted airport managing director and Terminal 5 team member John Stent.
Stent has taken over as managing director of the Terminal 5 project, which is due to go on site later this year. He was originally part of the project team in the mid-1990s before joining Stansted in 1997.
CrossRail chairman Sir Christopher Benson said Haste's was a key appointment. He said: "Norman Haste brings with him a wealth of experience in the planning and execution of major projects."
Haste's appointment comes as CrossRail, a joint venture between Transport for London and the Strategic Rail Authority, announced a shortlist of routes for the first phase of the scheme, which will run from east to west London.
The group said it was looking at options including routes to Heathrow, Stratford, Reading, Aylesbury and Watford Junction. CrossRail has £154m of government money to carry out feasibility work.
A final scheme will be selected for the first phase of CrossRail at the end of the year.