Moussaoui is the only person charged in the United States in connection with the September 11 attacks,
He also said he had lied about his involvement in the plot while he was in jail in August 2001, allowing the attacks to go ahead.
Prosecutors are seeking the death penalty for the self-confessed al-Qaeda operative on the grounds that his lies paved the way for the deaths of almost 3,000 people.
Moussaoui also told the court that British “shoe-bomber” Richard Reid was to join him as part of the crew in the White House suicide mission.
Reed was sentenced to life imprisonment after a failed attempt to blow up an American Airlines plane from Paris to Miami in December 2001.
Dressed in a green prison jumpsuit and white skull cap, his calmly delivered testimony held the courtroom spellbound, and was in stark contrast to previous outbursts.
His latest claim contradicts comments made last year when he pleaded guilty, when he said he was not supposed to be part of the September 11 hijackings but rather be in a second wave of al-Qaeda attacks and fly a plane into the White House.
Defence lawyer Gerald Zarkin asked Moussaoui: “Before your arrest, were you scheduled to be a pilot in the operation on September 11?”
“Yes,” said Moussaoui.
“I was supposed to pilot a plane to hit the White House,” Moussaoui added in his first public claim that he was part of the same plot as the other 19 hijackers.
When Mr Zarkin asked who told him to fly the plane into the building, Moussaoui replied: “Osama bin Laden.”
However shortly after his testimony, comments by operational mastermind Khalid Sheikh Mohammed contradicted the claim, saying in written statements that Moussaoui was meant to be part of the second wave of attacks.
Moussaoui, who testified for nearly three hours, is on trial to determine if he will be executed.
US District Judge Leonie Brinkema, who is hearing the case, said it could go to the jury as early as Wednesday.
Moussaoui – who does not talk with his court-appointed lawyers whom he has repeatedly tried to fire – had vowed to testify in the trial, though defence experts had said he would likely end up being one of the best witnesses for the prosecution.