The first Islamist terrorists to carry out their plan to murder on British soil without killing themselves in the process have been convicted of mutilating and attempting to behead a soldier, as the prime minister vowed to step up efforts to defeat the al-Qaida ideology of violence.

British-born Michael Adebolajo and Michael Adebowale were convicted of murdering Lee Rigby last May, after scouring an area around the Woolwich barracks hunting for a soldier to kill.

The savagery of the murder, in which Rigby, 25, was repeatedly stabbed and hacked at the neck by a cleaver, shattered community relations when mosques were attacked.

It was the first murderous attack in Britain by those motivated by the al-Qaida ideology of violence since the 7 July 2005 bombings of London's transport system by four suicide bombers.

It showed the continuing power of that ideology to turn Britons – in this case both with professional parents and who had been seemingly integrated into national life – into zealots who believed they were soldiers of Allah commanded to murder in retaliation for western foreign policy in Muslim lands.

Adebolajo, 29, the dominant one of the pair of converts to Islam, looked in the direction of the media and kissed a Qur'an as he was taken to the cells. The jury took just under 90 minutes to convict him and Adebowale after one of the most overwhelming cases of guilt in English criminal history, with key parts of the attack caught on CCTV and smartphones.

After running Rigby down with a car, mutilating him so badly that he had to be identified by dental records, and pulling his body into the road, they stayed at the scene and encouraged people to take pictures with their mobile phone cameras.

While convicting them of murder, the jury accepted their defence to a charge of attempting to murder armed officers who turned up to arrest them. Adebolajo and Adebowale claimed they charged at the armed officers clutching a machete and unloaded gun, not because they wanted to kill them but because they wanted to be shot dead and achieve martyrdom.

The harrowing and sometimes bizarre evidence was heard during the trial by Rigby's family, some of whom left the court as details of the death of the soldier who had survived Afghanistan, only to die on a London street, were heard by the Old Bailey.

The jury went out shortly after 11am yesterday, returning to deliver unanimous verdicts which were greeted by sobs. Rigby's stepfather, Ian, wiped away tears and put his arm around the soldier's mother, Lyn, who was crying.

After the verdicts, the Rigby family said in a statement their main focus was the soldier's two-year-old son: "These people have taken him away from us forever but his memory lives on in all of us and we will never forget him. We are very proud of Lee, who served his country, and we will now focus on building a future for his son Jack, making him as proud of Lee as we all are."

His wife, Rebecca, said: "This has been the toughest time of our lives and no one should have to go through what we have been through as a family. These people have taken away my baby's dad, but Lee's memory lives on through our son and we will never forget him."

Adebolajo and Adebowale, both from Christian Nigerian families, had attended events by extremist groups, led by Anjem Choudry, linked to the now banned al-Muhijaroun. Adebowale was attending such events last year while Adebolajo had led rallies several years ago.

The attack raised questions for MI5 and police, who had known of the men for eight years. Adebolajo has told those who have seen him since his arrest that repeated alleged attempts by MI5 to turn him into an informant had not played a part in the murder, which he blames purely on anger at western foreign policy. In 2010 Adebolajo had been arrested in Kenya, a gateway to terrorist areas in Somalia and it was claimed he was ill-treated with British complicity.

The security services also face questions over whether they underestimated the threat posed. The intelligence and security committee is investigating how much MI5 knew of the men's activities. The men are believed to have acted alone.

David Cameron said: "We have to redouble our efforts to confront the poisonous narrative of extremism and violence that lay behind this and make sure we do everything to beat it in our country."

Farooq Murad, secretary general of the Muslim Council of Britain, said: "Muslim communities then, as now, were united in their condemnation of this crime. This was a dishonourable act and no cause justifies cold-blooded murder."

Adebolajo, assessed by a psychiatrist as sane, was recorded at the scene brandishinga cleaver and a knife in his bloodied hands, and with the body of Rigby lying metres away, saying: "We swear by almighty Allah we will never stop fighting you. The only reason we have done this is because Muslims are dying every day. This British soldier is an eye for an eye, a tooth for a tooth."

In a police interview he said he had picked Rigby because he was the first soldier they saw. He was stabbed with knives including ones bought the day before from Argos. He claimed he slashed the neck because it was the most humane way to kill someone and added: "So I struck at the neck and attempted to remove the head."

Adebolajo told jurors that he was a "soldier of Allah" and was obliged to obey the command of Allah. He described how he held the soldier's hair as he hacked at his neck in a motion described by one witness as like a "butcher attacking a joint of meat".

Adebowale did not testify in his own defence and his police interviews were not heard by the court.

Mr Justice Sweeney, the trial judge, said he would pass sentence after a key ruling by the court of appeal in January on the use of whole-life terms. He is considering sentencing the men to die in jail. In two recent cases he rejected crown demands for whole-life sentences, including a terrorist case where a Muslim man was murdered by a Ukrainian man who also staged a bombing campaign, who received a minimum term of 40 years.