The Very Reverend Vivienne Faull said Britain "is not providing an adequate solution" to the current situation and said the cathedral is offering one of its vacant properties to a family.
On Friday Prime Minister David Cameron confirmed the UK will accept thousands more Syrian refugees, despite arguing that bringing peace and stability to the Middle East was the best solution to the crisis.
Mr Cameron has been under pressure after a photo emerged this week of the body of a 3-year-old boy refugee on a beach in Turkey.
Over 350,000 people have since signed a petition calling on government to do more.
Speaking during a visit to Portugal on Friday, Mr Cameron said he would set out details of the plans next week.
The PM said that Britain was already "providing sanctuary" to around 5,000 refugees from the camps, and had provided around £900 million in aid - more than any other European country.
Mr Cameron said: "We have already accepted around 5,000 Syrians, and we've introduced a specific resettlement scheme alongside those we already have to help those Syrian refugees particularly at risk.
"As I said earlier this week, we will accept thousands more under these existing schemes and we keep them under review. Given the scale of the crisis and the suffering of people, today I can announce that we will do more, providing resettlement for thousands more Syrian refugees.
"We will continue with our approach of taking them from the refugee camps. This provides them with a more direct and safe route to the United Kingdom, rather than risking the hazardous journey which has tragically cost so many their lives."
Mr Cameron said details of the scheme would be announced next week, after discussions with NGOs and other partners, and Britain would act with "our head and our heart".
It is understood York Minster will offer one of the residential properties it owns in the city to a family seeking refuge.
Dean Faull said: "Having seen the terrible suffering being experienced over recent weeks I urge the Government to change its stance and commit to welcoming more asylum seekers.
"I understand Britain is providing considerable help through its overseas aid budget but it is now clear that this is not providing an adequate solution and we need to do more to help those in desperate need."
The dean pointed to York's history of providing sanctuary for persecuted people as she urged more in the city to come forward with help.
She said: "York has a long history of offering sanctuary and I would welcome the chance to convene a meeting so we can respond to this crisis as a city and commit to welcoming some families.
"As part of this, York Minster offers a refugee family accommodation in one of its vacant properties."