Shadow chancellor John McDonnell has apologised "from the bottom of my heart" for the offence caused by calling for Irish republican terrorists to be honoured but suggested his comments may have helped the peace process.
The senior Labour MP, appointed by Jeremy Corbyn to lead the party's economic policy, said it had been a "mistake" to use the words and accepted he had "clearly" caused offence.
David Cameron said Mr McDonnell should be "ashamed" of the comments when the issue was raised during Mr Corbyn's first session of Prime Minister's Questions as opposition leader.
The row follows Mr McDonnell's remarks in 2003 that "the bombs and bullets and sacrifice" of the IRA had brought Britain to the negotiating table.
Challenged about the comments on BBC1's Question Time Mr McDonnell said: "I accept it was a mistake to use those words, but actually if it contributed towards saving one life, or preventing someone else being maimed, it was worth doing because we did hold onto the peace process.
"There was a real risk of the republican movement splitting, and some continuing with the armed process. If I gave offence, and I clearly have, from the bottom of my heart I apologise."
In 2003 Mr McDonnell told a meeting in London: "It's about time we started honouring those people involved in the armed struggle. It was the bombs and bullets and sacrifice made by the likes of (hunger striker) Bobby Sands that brought Britain to the negotiating table.
"The peace we have now is due to the action of the IRA."
During the show on Thursday night Mr McDonnell, who had also described the presence of British troops in Northern Ireland as an "occupation" said he had to talk to republicans on their terms.
The shadow chancellor insisted "I reject political violence" and added: "I had to use the language that republicans understood so we could secure the path to peace.
"There were risks but it was worth taking because now people are not dying on the streets of Northern Ireland."
Following Mr McDonnell's apology Nigel Dodds, the DUP's leader at Westminster, said it is crucial that his apology is a meaningful one, adding that he should not try to justify his words.
"John McDonnell's attempt to address this issue is welcome and long overdue," said Mr Dodds.
"This should not have been dragged out of him by entirely justified public outrage only once these appalling remarks were brought to general attention."
The shadow chancellor used his Question Time appearance to also apologise for an "appalling joke" in 2010 about wanting to assassinate Margaret Thatcher.
"It was an appalling joke. It's ended my career in stand-up, let's put it that way, and I apologise for it as well."
The shadow chancellor said Labour favoured a return to the "reasonable" 50p top rate of income tax, playing down suggestions that he and Mr Corbyn could push for it to rise to as much as 70p.
He also insisted that Labour was not advocating withdrawal from the Nato alliance.
Mr McDonnell also said he had spoken to his party leader about the row following Mr Corbyn's decision not to sing the national anthem at a Battle of Britain memorial service.
"I said afterwards 'why didn't you sing?' and he said 'actually I normally do', but it was quite a moving event and he was casting his mind back to the war," Mr McDonnell said.
"The national anthem isn't just for those who are monarchists, it's for everyone and it represents the whole country and that's why people sing it."
Meanwhile, Mr Corbyn has given his clearest indication yet that Labour will campaign in favour of continued European Union membership in the in/out referendum promised by David Cameron by the end of 2017.
Writing in the Financial Times, the new Labour leader - who had been criticised for appearing to leave the door open for backing withdrawal - said: "Labour is clear that we should remain in the EU. But we too want to see reform."
While the party opposes reforms being sought by Mr Cameron that would reduce workers' rights, the shadow cabinet was in agreement that the answer was "not to leave the EU but to pledge to reverse those changes with a Labour government elected in 2020", Mr Corbyn said.
His comments came as Mr Cameron told business leaders that Labour would get "nowhere near power" under Corbyn's leadership.
Speaking at a business reception in London, the Prime Minister described the new Labour leader's support for higher taxes and nationalisation as a "throwback" to the 1980s, adding: "I think the British people have moved a long way from that.
"No-one wants to go back to those ideas."
Liberal Democrat leader Tim Farron raised the prospect of defections to his party, telling the Evening Standard he had received "unsolicited texts" from well-known Labour figures "distressed" about the direction Mr Corbyn was taking them.