The Ryanair plane skidded on to the grass moments after landing just after 9am.
No one was injured, police said.
A Strathclyde Police spokesman said: “It happened just after 9am as it landed. It slipped on to the grass. No one was injured and all the passengers have been taken to the terminal.”
Alex Paton, a passenger on the plane, said: “Everyone was pretty shocked. I don’t think most people realised we were in a field.
“It wasn’t until we were off that we realised how deep the wheels had gone into the field. Don’t get me wrong, it didn’t go into the field that long before it stopped.
“It was almost like the back end of the plane slid out a bit.”
He added: “I think the original plan was to pull the plane out of the field but, when you see it, there’s no way you’re pulling that plane out easily.”
Passenger Philip O'Reilly, 50, said: "We landed on the tarmac, the plane was moving very fast, and I realised it wasn't going to stop in time.
"The pilot, as far as I know, reversed the thrusters and either steered off the runway or accidentally came off, I don't know which."
Mr O'Reilly, from Castlebar, County Mayo, Ireland, went on: "All I can say is it was very skilful - he did a great job.
"Everybody came off the plane without our hand luggage, and we all stood around in freezing cold.
"The fire brigade went into the plane, checked everything out and brought our luggage out to us, and we got three buses back to the airport."
Mr O'Reilly, who was flying to Scotland to visit relatives in Aberdeen, said: "It was very exciting - nobody was hurt, so everybody was happy."
He went on: "We realised straightaway what had happened.
"It just came to a sudden stop and the front wheels plunged into the grass verge.
"It's going to take a long time to move that plane, that's for sure."
An airport spokesman said the plane was an incoming Ryanair flight.
He said: “All I know at the moment is that after it landed it hit a patch of ice after completing its landing.”
The aircraft was not damaged, and all passengers were taken off the plane unharmed, he said.
Strathclyde Fire and Rescue were at the scene as a precaution.
Ryanair said the plane involved was an incoming flight from Dublin, with six crew on board.
The airline would not say how many passengers were on board the Boeing 737-800, which can carry up to 189 travellers.
The airline said: “After a normal landing was completed and while taxiing from the runway the aircraft encountered ice and slid just off the runway on to the grass verge.
“Passengers disembarked normally and were bussed to the terminal.
“Ryanair engineers are at the aircraft, which appears to have suffered no damage and they are working with Glasgow Prestwick to return the aircraft to the stand area so that the runway can be reopened with minimum delay.”
The statement added: “Ryanair apologises for any disruption suffered by passengers as a result of this incident.”
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