The prime minister claimed he would win what is expected to be a knife-edge Commons vote on Saturday.
"This is our chance in the UK as democrats to get Brexit done, and come out on 31 October," he said.
The DUP is against concessions he made to the EU on customs checks at the point of entry into Northern Ireland.
The party's deputy leader, Nigel Dodds, accused the prime minister of being "too eager by far to get a deal at any cost".
The PM must also convince Tory Brexiteers, 23 independent former Tory MPs and Labour MPs concerned about protection for workers and the environment in the new deal.
He denied he would meet the same fate as his predecessor Theresa May, who repeatedly failed to get a Brexit deal through Parliament.
"I am very confident that when my colleagues in Parliament study this agreement that they will want to vote for it on Saturday and in succeeding days," he said at an EU summit in Brussels.
Appealing to the DUP, which the government relies on for support in key Commons votes, he insisted the UK could leave the EU "as one United Kingdom" and "decide our future together".
Mr Dodds earlier said he expected a "massive vote" against Mr Johnson's deal on Saturday in the House of Commons - and the DUP expected to "play a crucial role" in amending the legislation.
The new deal is largely the same as the one agreed by Theresa May last year - but it removes the controversial backstop clause, which critics say could have kept the UK tied indefinitely to EU customs rules.
Northern Ireland would now remain in the UK's customs union, but there would also be customs checks on some goods passing through en route to Ireland and the EU single market.
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