Boaz Trust, which supports asylum seekers, made the plea as the European Union announced it was to boost the centres in Greece and along the Balkans migration route to shelter 100,000 more people as winter loomed.
EU nations also agreed to expand border operations and make full use of biometric data like fingerprints as they register and screen migrants, before deciding whether to grant them asylum or send them home.
"The immediate imperative is to provide shelter," European Commission president Jean-Claude Juncker said after chairing the often bad-tempered mini-summit of 11 regional leaders in Brussels.
"It cannot be that in the Europe of 2015 people are left to fend for themselves, sleeping in fields."
But Dave Smith from the Boaz Trust said there "has to be financial support and there has to be all sort of other support for those countries where they are coming in".
He added: "You can't just leave them there, that's just unfair."
Meanwhile the Slovenian Prime Minister Miro Cerar said his small Alpine nation was being overwhelmed by the refugees and was not receiving enough help from its EU partners.
He put the challenge in simple terms: if no fresh approach is forthcoming "in the next few days and weeks, I do believe that the European Union and Europe as a whole will start to fall apart".
The leaders agreed to rapidly dispatch 400 border guards to Slovenia as a short-term measure.
But Mr Smith told Premier he was concerned that Turkey, a country that many say is another key source of the flow, was not invited to the EU talks.
"They should have been in on it," he said.