Michael Adebolajo, 28, and Michael Adebowale, 22, are accused of murdering 25-year-old Lee Rigby as he walked to his barracks in Woolwich, south London, in May.
The two defendants, both Britons of Nigerian descent, deny murder.
As the trial opened at London's Old Bailey court, prosecutor Richard Whittam said the pair had run Rigby over with a car, rendering him unconscious, before attacking him with the knives in a "cowardly and callous murder".
Rigby's relatives watched from the courtroom as Whittam told the jury of eight women and four men that Adebowale had stabbed and cut the soldier while Adebolajo tried to cut his head off.
They had dragged his body into the middle of the road, Whittam said, because "they wanted members of the public to see the consequence of what can only be described as their barbarous acts".
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"They had committed, you may think, a cowardly and callous murder by deliberately attacking an unarmed man in plain clothes from behind, using a vehicle as a weapon, and then they murdered him and mutilated his body with that meat cleaver and knives."
He added: "They both attacked the motionless body of Lee Rigby. He was repeatedly stabbed and it appears it was Michael Adebolajo, the first defendant, who made a serious and almost successful attempt to decapitate Lee Rigby with multiple blows to his neck made with the meat cleaver.
"At the same time as Michael Adebolajo used the meat cleaver, Michael Adebowale was using a knife to stab and cut at Lee Rigby's body."
When the police arrived, Adebolajo ran at a police car with the meat cleaver raised and was shot by officers, the prosecutor said.
Adebowale ran at police with a gun, and was also shot.
Adebolajo and Adebowale are further accused of attempting to murder a police officer, and conspiracy to murder a police officer.
The murder took place in the middle of the day as a group of school children were returning from a local library, Whittam said.
Members of the public turned them back to avoid the "awful" scene, he added.
Adebolajo has asked to be known as Mujaahid Abu Hamza in court, while Adebowale wants to be called Ismail Ibn Abdullah.
