Emergency officials helped 127 passengers and five crew off the plane just after 11:00 local time, but no one was seriously injured.
Snow and freezing rain has been falling from Texas to New England over the past several hours.
Schools, businesses, and the US government have closed as a result.
Pictures from LaGuardia airport show the plane, a Delta MD-88, resting on an embankment having pushed through a fence.
“That runway had been ploughed literally minutes before, and other pilots had reported good braking action,” New York and New Jersey Port Authority Director Patrick Foye told reporters.
The flight, Delta 1086, was attempting to land at LaGuardia after flying from Atlanta. It veered to the left shortly after making contact with the runway, but avoided crashing into nearby Flushing Bay.
Two passengers were transported to a hospital, but no serious injuries have been reported.
At one point the plane was leaking fuel, but emergency responders were able to stop the leak.
The National Transportation Safety Board (NTSB) has been dispatched to the scene, Patrick Foye said.
The airport has been closed, and is expected to reopen at 19:00 local time, Federal Aviation Administration (FAA) says. Planes en route to the LaGuardia have been diverted to nearby airports.
LaGuardia is one of the most difficult airports to land at in the US owing to its close proximity to three other busy airports.
Elsewhere:
The snow is expected to largely skip Boston, which needs just two more inches of snow to break a record set during the 1995-1996 season.
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