With only days to go until MPs vote on Theresa May's deal, the foreign secretary issued a stern rebuke to Brussels, saying they needed to be "flexible in the negotiations" and accept that the UK has a "very clear ask" on the changes needed to get the deal over the line.
The prime minister will use a speech in Grimsby to appeal to the EU to give ground on the divisive Irish backstop but the mood among her top team is grim, as attempts to secure tweaks to her deal have floundered in recent days.
"This is a moment of change in our relationship between the UK and the EU and history will judge both sides very badly if we get this wrong," Mr Hunt told the Today programme.
"We want to remain the best of friends with the EU. That means getting this agreement through in a way that doesn't inject poison into our relations for many years to come.
"That's what the UK has said we want to do, it's what most people in the UK want and feel very strongly about.
"But it does need the EU also to be flexible in these negotiations and understand that we now have a very, very clear ask."
MPs will vote on Tuesday on whether to support Ms May's deal, in a second meaningful vote that could determine the course of Brexit.
It follows the humiliating 230-vote defeat Ms May suffered the last time the Commons passed judgment on her Brexit deal in January.
Mr Hunt said: "We know what it would take to get a deal through the House of Commons, and that is for a significant change to allow the attorney general to change his advice to the government and say we couldn't be trapped in a customs union forever.
"That's not an unreasonable thing to ask and we have made, I think, some progress in the last few days. There's a bit more to make. It's entirely possible to get there.
"And frankly I think future generations, if this ends in acrimony, will say that the EU got this moment wrong. And I really hope they don't."
Conservative Party deputy chairman James Cleverly said it was inevitable that the UK and the EU have ended up at this point and urged colleagues to hold their nerve as "the final point of a negotiation of this magnitude it gets most intense, it gets most difficult, it gets most challenging".
He added: "I think we are getting close to the point where modest but significant changes can unlock this."
Elsewhere, international trade secretary Liam Fox urged Conservative Brexiteers to rally behind the deal or run the risk of Brexit not happening.
"The thing that I fear is that there will be ... a risk that we might not deliver Brexit at all," he told BBC's Newsnight.
"In parliament there are a large number of MPs who do not see it as their primary objective to deliver on the referendum and would want to keep us locked to the European Union."
Negotiators are preparing to work through the weekend in a frantic effort to break the deadlock over the backstop measures, which are aimed at preventing a hard border with Ireland if no alternative trading arrangements are in place.