Sir Keir urged the Prime Minister to call a general election, as he accused the Government of failing to heed warnings which led to a series of crises for ministers over the last week.
But Mr Sunak hit back, claiming the Labour leader cannot be trusted due to his “principles-free, conviction-free type of leadership”.
At Prime Minister’s Questions on Wednesday, Sir Keir told MPs: “Probation, prison, schools, China… yet again Inaction Man fails to heed the warning and then blames everyone else for the consequences.”
In recent weeks, Mr Sunak’s Government has had to deal with the escape of former solider Daniel Khalife from Wandsworth Prison, crumbling concrete in school buildings, and reports of Chinese espionage in Westminster.
The Prime Minister became personally embroiled in the row about reinforced autoclaved aerated concrete (Raac) after suggestions that, when chancellor, he approved 50 schools to be rebuilt per year, rejecting an application for 200 to be given the same treatment.
Pressing Mr Sunak, Sir Keir said: “On Sunday, the Home Secretary celebrated her first anniversary in post – that is if you overlook the six days she missed when she was deemed a national security risk.
“In that year, 40,000 people have crossed the Channel on a small boat, and the taxpayer is now spending £6 million a day on hotel bills.
“He is failing to stop terrorists strolling out of prison, failing to guard Britain against hostile actors, he has completely failed to stop the boats. How can anyone trust him to protect the country?”
Mr Sunak responded by accusing Labour of leading plans to block housing reforms in the Government’s Levelling Up and Regeneration Bill, which is currently being considered in the House of Lords.
The Prime Minister said: “He talks about trust, he tried in this House to talk the talk on housebuilding, but at the first sign of a cheap political hit, what did he do? He has caved in.
“Rather than make the right long-term decisions for the country, he has taken the easy way out. It is typical of the principles-free, conviction-free type of leadership that he offers.
“Flip-flopping from being a builder to a blocker. The British people can’t trust a word he says.”
Calling for a general election, Sir Keir went on: “No-one voted for this shambles. No-one voted for him. So how much more damage do the British public have to put up with before he finally finds the stomach to give them a say?”
But the Prime Minister responded: “We are getting on for the British public, just in the last week announcing a new landmark deal for British scientists, attracting £600 million for new investment for our world-leading auto industry, and wages now rising at the fastest rate on record.
“Where has he been this week? Locked away with Labour’s union paymasters, promising to give them more power and scrap the laws that protect British families and their access to public services.
“It is clear it is only the Conservatives that are on the side of the hardworking British public.”
The Labour leader had earlier called for a “full audit of UK-China relations”, following reports of Chinese spying in Westminster.
He also pressed the Prime Minister to reveal if Foreign Secretary James Cleverly had spoken with his Chinese counterpart about the allegations when he visited Beijing in August, before they became public knowledge.
He said: “When I asked the Prime Minister on Monday whether the Foreign Secretary raised the specific issue of the alleged spy arrested in March when he visited China a few weeks ago, the Prime Minister would only say that he raised that ‘type of activity’ but avoided specifics.
“So, I ask the Prime Minister again, did the Foreign Secretary raise this specific case when he visited China, yes or no?”
But Mr Sunak stopped short of giving an explicit answer, telling MPs: “I refer (Sir Keir) to my previous answer where I said very clearly that the Foreign Secretary raised these issues with the Chinese foreign minister who he met, as did I when I had my meeting with premier Li (Qiang) over the weekend.”