He insisted in parliament that the prime minister had "secured legally binding changes that strengthen and improve the withdrawal agreement and political declaration".
She declined to appear at an urgent question called by the Labour leader, Jeremy Corbyn, who said parliament would not tolerate further delay.
European Commission president Jean-Claude Juncker said: 'Our agreement provides meaningful clarifications and legal guarantees to the Withdrawal Agreement and backstop.
Tajani said Britain's departure from the EU would deter other countries from leaving the bloc, adding: "We need to change the European Union but we need to stick together".
He added: "It is in this spirit that today the Prime Minister and I have agreed on a joint legally-binding instrument relating to the Withdrawal Agreement".
"It's a matter now of avoiding the biggest mistake of all - a chaotic Brexit without contractual arrangements in place", Tajani said in an interview due to be published on Saturday.
MPs fear a no-deal Brexit would unleash chaos at the border and cause food and medicine shortages.
Lidington said that "negotiations are continuing" in Strasbourg, and the British government will update Parliament at the "earliest opportunity".
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Ambani's performance was enhanced by the musical fountain and light show at the newly-inaugurated Dhirubhai Ambani Square. Moritz, Switzerland (no less!) where the performers included Coldplay's Chris Martin and the band The Chainsmokers.
The prime minister was warned that her position could be in jeopardy unless she stuck to commitments she made about this week's votes.
The meaningful vote on the PM's Brexit deal will happen on Tuesday, but it is widely anticipated MPs will reject it.
And senior Labour backbencher Yvette Cooper said the PM would be guilty of a "straight-up lie" if she failed to go through with votes allowing MPs to delay Brexit.
He added: "I think a meaningful vote with an addendum saying this House will support a deal if such and such is done might be a way of uniting the party or limiting the scale of the defeat".
Talks will resume on Monday morning with the aim of securing changes to the deal before Tuesday's vote.
A leading cabinet source said there was increasing concern about the prospect of a no-deal Brexit by accident - on 29 March or at the end of an extension period - if there was no parliamentary majority for May's agreement because time was running out to pass legislation necessary to block one.
Environment Secretary Michael Gove - one of the leaders of the official Leave campaign in the referendum - became the latest Cabinet minister to urge MPs to vote for Mrs May's deal.
May's apparent breakthrough came after a day of drama on both sides of the channel, with many centrist Tories outraged by the suggestion that she had been planning to pull her meaningful vote and replace it with a provisional one showing the European Union what Eurosceptics in her party would accept.