The plan will "fundamentally transform" the hub, giving New York the "world-class airport" it deserves, Gov. Andrew Cuomo said in a statement.
"This represents a major milestone for the project — the first of many — and I look forward to seeing construction begin later this year," he said.
Two-thirds of the $4 billion project will be funded by the private consortium LaGuardia Gateway Partners, which will build, operate and maintain the airport, officials said.
If the project goes over budget, or is delayed, it will be handled by the LGP — not the Port Authority, Cuomo's office said.
Construction is set to begin this year and includes a gut renovation of Terminal B, creating a central hall that connects the terminals.
The redesign also includes a plan to provide more taxi space by moving terminals closer to the Grand Central Parkway, as well as new roads and parking.
Pat Foye, the Executive Director of the Port Authority, said the private-public deal "shifts the risks of complex construction in an operating airport environment from the Port Authority to well-capitalized private parties."
"Approval of the lease with LGP is both the first major step toward Gov. Cuomo's modernization of LaGuardia Airport and the largest public-private partnership for new transportation infrastructure in U.S. history," he said in a statement.