There seems to be air inside the Death Star tubes (going by the nature of some of explosion they have to out run).
So you could have the fire extinguished the moment the damaged ship hits the vacuum of space.
Another possibility could be that Lando ejects the cockpit section of the Falcon as it bursts out of the vent (it could have little thrusters and act like a giant escape pod).
That would mean Han's prediction about not seeing the ship again has come true, but Lando still lives.