Passport Office chief executive Paul Pugh has apologised for any "distress" caused by a large backlog in applications.
Mr Pugh said he had considered standing down over the crisis, but told MPs: "It is my job to lead the agency and that is what I intend to do."
He admitted the agency had not done enough to cater for a surge in demand for passports.
The PCS union has not ruled out strike action over the crisis.
Last year, Mr Pugh estimated applications would surge by as much as 350,000 this summer because of overseas embassies shutting their passport desks and transferring operations to Britain.
Pressed on why there had still been delays in processing passports, he told the Home Affairs Committee the actual surge had turned out to be between 390.000 and 400,000, although that was "over the whole 12 month period".
He insisted adequate preparations had been made for the closure of the overseas centres and other factors may have caught the Passport Office out, such as a "significant shift in the seasonality of customers' behaviour".
He said he had ordered an independent review into the agency's forecasting system.
'Anger and distress'
He confirmed figures provided earlier to the Committee by Mike Jones, Home Office group secretary at the Public and Commercial Services Union (PCS), that as of Wednesday around 480,000 passport applications were logged as work in progress at the agency - nearly double the number at the same time last year.
Mr Jones earlier revealed that this figure has surged from around 290,000 work-in-progress applications in March this year.
What about that picture?
The Home Office had failed to admit there was a problem at the Passport Office until this picture - showing paperwork piling up in a side office - had been leaked to the Guardian newspaper, said Labour MP and committee member Yasmin Qureshi.
Surely, instead of conducting a "witch hunt" to find the culprit, she said, they should be congratulated for performing a public service.
Paul Pugh said all staff had to abide by civil service rules on unauthorised contact with the media, but added: "What the picture demonstrated, I think, was a well-ordered, well-contained area for holding work ready for consideration and processing."
The Passport Office has been accused of putting thousands of summer holidays in jeopardy after it emerged interviews for first-time applicants were suspended in London as staff struggled to deal with the backlog.
Mr Vaz asked Mr Pugh if he would like to apologise "to all those who have been waiting".
Mr Pugh replied: "I absolutely recognise the anger and distress that some people have suffered and I would like to put on record that yes, in every case where we haven't met our service standards, where we haven't been able to meet the customer's needs, yes, certainly, we are sorry for that."
Mr Vaz asked: "We can take that as an apology?"
Mr Pugh replied: "It is an apology."
Mr Vaz revealed he had sent a text message to the Home Secretary, Theresa May, over the weekend in a bid to resolve a passport crisis faced by one of his constituents in Leicester.
The committee chairman explained that the woman, who was set to travel on Wednesday, had made the journey to the passport office in Durham, from Leicester, to discover her passport was not ready.
After failing to get an answer from Mr Pugh at the Passport Office, Mr Vaz contacted the home secretary and ultimately the constituent was able to travel.
Mr Vaz ordered Mr Pugh to appear before the committee again in a month's time, by which time, he warned, the work-in-progress figures must have been reduced to the public's satisfaction.
The Passport Office boss claimed that in 2010 staff at the agency were so "under-employed" they were filling in time by "tidying the office" and "in some cases reading books".
But he denied union claims that the closure of 22 passport centres and the loss of 550 jobs - as well as the closure of the overseas units - had caused the crisis.
Mr Jones said Passport Office managers had decided to "use massive amounts of overtime", rather than extra staff, to deal with the extra workload, which had "burned the staff out".
He said Mr Pugh had ignored requests for talks on pay, staffing and "an end to privatisation" - and he refused to rule out a strike ballot if their calls for negotiations "continue to be ignored".
Mr Pugh, who earns £104,000 a year, said the PCS was not the only voice of workers at the Passport Office but he had no objection to holding talks with with its representatives, including Mr Jones.
After weeks of mounting public anger, Home Secretary Theresa May announced a raft of measures aimed at clearing the backlog.
Fast-track processing fees for passport applicants who need to travel abroad urgently have been dropped, and people renewing their UK passports from overseas are being given a 12-month extension to their existing passport.
Those applying for passports overseas on behalf of their children will be given emergency travel documents.