The taoiseach said he believed Mr Johnson would agree to keep the UK closely aligned to European rules on labour rights and environmental protections to ensure he can sign a tariff-free and quota-free trade with Brussels.
The prime minister has vowed not to extend the transition period beyond December 2020, giving himself only 11 months to negotiate the future trading relationship with Brussels or risk crashing out of the bloc.
Arriving at an EU summit, Mr Varadkar said he was keen to strike a “trade deal plus” with the UK and talks with the prime minister led him to believe that Mr Johnson was ready to accept some of the EU’s red lines.
Mr Varadkar said he was “relieved” for the UK and for Ireland that the “deadlock and gridlock” was over and called for the Brexit deal to be ratified as soon as possible.
Asked about the next stage of the Brexit process, he said: “I’d like to ensure we still have tariff-free and quota-free trade between Britain and the EU and that there is a set of minimum standards so that nobody feels that there’s unfair competition or anyone’s trying to undercut them when it comes to labour rights, environmental protection and issues like that.
“My conversations with Mr Johnson, I think he’s probably in a similar space, so it’s a case of now getting on with it. I think the prospect is there to develop a new, future economic partnership between the EU and UK and one that’s going to be mighty and one that is going to be good for all of us.”
The outright Conservative majority will make it easier for the prime minister to secure a trade deal, despite the tight timetable, Mr Varadkar said.
He said: “I think striking a trade deal by the end of 2020 is enormously ambitious, but we won’t achieve if we don’t try and we’re determined to try. I think the fact prime minister Johnson has a clear majority makes a big difference.
“I think the prime minister’s hand is strengthened. It’s always easier for us to negotiate with a partner who has a strong personal mandate in the parliament.”
European Council president Charles Michel said Brussels wanted to see the Brexit deal to be ratified “as soon as possible”, as he sent his congratulations to the prime minister.
Mr Michel said the EU was “ready for the next phase”, adding: “We will negotiate a future trade deal which ensures a true level playing field”.
Asked whether a deal with the UK can be sealed in the next 12 months, Mr Michel said: “Regarding Brexit, it’s not my intention to predict based on the experience of the past, but I hope that we will be able to work very hard with the British government.”
A string of populist leaders sent their best wishes to Mr Johnson, with Italy’s right-wing opposition leader Matteo Salvini tweeting: “Go Boris Go!”
Marine Le Pen, who heads France’s National Rally party, congratulated Mr Johnson in her native language for his “overwhelming victory”.
Bulgaria’s centre-right prime minister Boyko Borissov offered congratulations to Mr Johnson and his party.
“I believe that we will continue to work well together and maintain high-level co-operation between Bulgaria and the UK,” Mr Borissov tweeted.
Guy Verhofstadt, the outspoken European parliament Brexit negotiator, put out a statement accepting that the UK was on course to leave the bloc.
He said: “Brexit will now happen. The British people have confirmed their referendum decision of 2016. The EU must now focus on building a new close, fair and lasting partnership with Britain. It is in our common interest.”