The House of Lords Committee on Charities said innovative small charities could be the worst hit as they are already under financial pressure.
It said public bodies must recognise their importance with grants to test new ideas and by avoiding overly-prescriptive contracts for services.
Charities receive the EU money from structural and investment funds, in particular the European social fund (ESF), and must get more support with funding to meet core costs, the peers said.
In evidence to the committee, the Royal Mencap Society described the ESF as a "major source of revenue" to help raise living standards and provide job opportunities for young people and the long-term unemployed.
Medical research charities raised concerns about losing EU funding and opportunities for collaboration after Brexit, such as through the Horizon 2020 innovation programme.
Civil society minister Rob Wilson told the committee the Government was listening to charities' concerns and said if they can show "strong value for money" they will continue to be funded.
But the peers called on the Office for Civil Society to audit the potential impact of Brexit on charities, including the loss of funding and research collaboration, and publish a report by the end of the year.
Elsewhere, the committee said it had "grave concerns" about the Charity Commission's proposal to charge charities an annual registration fee.
Peers said any such plan must have evidence to back it up and the Commission must show how any resulting improvements to its regulation of the sector would directly benefit charities.
The committee also urged ministers to commission charitable services based on impact and social value rather than simply going for those that cost the least.
And it criticised the Government for causing "unnecessary concern" with an "anti-advocacy" clause in grant awards and with 2014 laws which "threatened the vital advocacy role of charities", and urged ministers to improve the way it consults the sector when bringing forward policies.
Committee chairman Baroness Pitkeathley said: "Charities are the lifeblood of society. They play a fundamental role in our civil life and do so despite facing a multitude of challenges. Yet for them to continue to flourish, it is clear that they must be supported and promoted.
"We found that charities lead the way with innovation, but that this is at risk of being stifled by the 'contract culture'. And while advocacy is a sign of a healthy democracy, and is a central part of charities' role, this role has been threatened by Government.
"We hope that charities will be encouraged by this report; that the Government will respect their role; and that in addition it will value the connections charities have with all sections of society, and encourage the vital scrutiny they provide."
Sir Stuart Etherington, chief executive of the National Council for Voluntary Organisations, seized on the committee's concerns over the Commission's plans to charge charities a fee.
"We know the Charity Commission is under financial pressure, and that it is in charities' interests to have a well-funded regulator," he said.
"But the Lords have been unambiguous in saying that they do not think the Charity Commission have properly considered their plans to levy fees on charities.
"The Commission must now consider the questions of principle that the committee have raised before it launches into consultation on the detail of a charging scheme."
He added: "Given concerns over partisan appointments to the Charity Commission board, there is a clear case that any move to charging should go hand in hand with an overhaul of the Commission's governance to ensure it is properly independent of the government of the day."
The Local Government Association urged charities to engage with devolution of powers to councils as it will give them "considerable" opportunities.
A spokesman said: "While most of the deals agreed so far have focused on driving economic growth, councils have been consistently clear that devolution needs to go further and support local public service reform to better tackle issues such as long term unemployment, youth offending and homelessness.
"Charities and the voluntary sector often play a vital role in providing services that address the needs of specific communities and in helping drive increased levels of participation.
"As devolution provides local areas with greater freedom over the design of public services we expect they will continue to develop their role in shaping effective local public services."