James Cleverly has described the Government’s Rwanda deportation plan passing through Parliament as a “landmark moment in our plan to stop the boats”.

The House of Lords had been engaged in an extended tussle over the Safety of Rwanda (Asylum and Immigration) Bill on Monday, sending it back to the Commons five times in a bid to secure changes.

The unelected chamber ended the deadlock after MPs rejected a requirement that Rwanda could not be treated as safe until the secretary of state, having consulted an independent monitoring body, made a statement to Parliament to that effect.

The Government said the Lords amendment was “almost identical” to the previous ones overturned by MPs.

In a video posted to social media early on Tuesday, the Home Secretary said the Bill “will become law within days”.

He said: “The Act will prevent people from abusing the law by using false human rights claims to block removals. And it makes clear that the UK Parliament is sovereign, giving Government the power to reject interim blocking measures imposed by European courts.

“I promised to do what was necessary to clear the path for the first flight. That’s what we have done.

“Now we’re working day in and day out to get flights off the ground.”

Earlier in the upper House, the opposition did not press its demand for the Bill to include an exemption from removal for Afghan nationals who assisted British troops after what critics hailed as a concession.

A Home Office minister said the Government will not send those who are eligible under the Afghan Relocations and Assistance Policy (Arap) to Rwanda.

The new law aims to clear the way to send asylum seekers who cross the Channel in small boats on a one-way flight to Kigali.

The legislation and a treaty with Rwanda are intended to prevent further legal challenges to the stalled scheme after the Supreme Court ruled the plan was unlawful.

As well as compelling judges to regard the east African country as safe, it would give ministers the power to ignore emergency injunctions.

The Prime Minister says the policy will act as a deterrent to migrants attempting to make the perilous journey across the world’s busiest shipping lane.

In a statement, shadow home secretary Yvette Cooper called the Bill “an extortionately expensive gimmick rather than a serious plan to tackle dangerous boat crossings”.

“The Rwanda scheme will cost more than half a billion pounds for just 300 people, less than one per cent of asylum seekers here in the UK – and there is no plan for the 99 per cent,” she said.

“Instead of spending £2 million per asylum seeker on this failing scheme they should be putting that money into boosting our border security instead – that is Labour’s practical plan.

“This is the third new law the Tories have passed on Channel crossings in two years – each one has made the chaos worse and even senior Tory MPs don’t believe this third law will work.

“As former immigration minister Robert Jenrick has said this is just a plan to get a few symbolic flights off before an election. Now the new law has passed, the Conservatives will immediately sign another £50 million cheque to Rwanda on top of the £200 million sent so far, even though not a single asylum seeker has yet been sent.

“The Conservatives should drop this eye-wateringly expensive election stunt and instead adopt Labour’s practical plan boost our border security with new cross-border police and new counter terror style powers to crack down on the criminal gangs, and a new Returns and Enforcement Unit to remove those who have no right to be here.”

Earlier on Monday, Rishi Sunak blamed Labour peers for holding up the Bill, as he acknowledged he will miss his self-imposed spring target for getting the Rwanda scheme off the ground.

Speaking in the Lords, leading lawyer and independent crossbencher Lord Anderson of Ipswich, who has been a prominent critic of the legislation, said: “The purpose of ping-pong is to persuade the Government through force of argument to come to the table and agree a compromise. They have refused pointedly to do so.”

Green Party peer Baroness Bennett of Manor Castle hit out at the “dreadful, international law-busting Bill”.

In an unusual intervention as the Bill cleared Parliament shortly after midnight, the Tory Lords leader Lord True said: “Of course this House has a major role, an abiding role in asking the elected House to think again.”

But he added: “I think it is important that we have a discussion about what are the limits and what is the place of the House in scrutinising and indeed challenging legislation put forward.

“I think in the weeks and months ahead we do need to reflect whether four or five times sending something back to the elected House is the best way to enable the King’s Government to be carried on.”

Responding to critics of the legislation, Home Office minister Lord Sharpe of Epsom said: “This Bill does comply with international law, it is profoundly moral and patriotic to defend the integrity of our borders, it’s profoundly moral and patriotic to prevent needless loss of life in the Channel and to put the criminal gangs out of business.”

Having completed its parliamentary passage, the Bill now goes for royal assent.