Mr Odinga, who was defeated by incumbent Uhuru Kenyatta, has accused the electoral authorities of fraud tantamount to "democracy's slaughter".
But foreign observers described the election as free and fair.
The international community has urged calm amid fears that the fraud claims could result in violence.
The official results gave President Kenyatta about 54% of the vote, and Mr Odinga 45%.
Mr Odinga, who had previously refused a court appeal against the outcome, said the opposition had collected concrete evidence of massive irregularities that were allegedly used to rig the vote.
"We have now decided to move to the Supreme Court to lay before the world the making of a computer-generated leadership," he told journalists.
European Union foreign affairs chief Federica Mogherini and former UN Secretary General Kofi Annan were among those who had urged Mr Odinga to seek redress through the courts. But he insisted that his decision did not mean he was giving up the fight.
"By going to court we aren't legitimising misplaced calls from some observers for us to concede but are seeking to give to those who braved the long lines in the morning chill and hot afternoon... a chance to be heard."
He also repeated his call for peaceful protests, saying Kenyans had the constitutional rights to demonstrate, strike and carry out acts of civil disobedience.
But on Monday, Mr Odinga's strike request was largely ignored.
"This is just the beginning, we will not accept and move on," he added.
In 2013, Mr Odinga challenged the presidential results in the Supreme Court and lost. Now, he said, he the court had an opportunity to demonstrate its independence.
"Our decision to go to court constitutes a second chance for the Supreme Court. The court can use this chance to redeem itself or, like in 2013, it can compound the problems we face as a country."
Kenya's opposition leader Raila Odinga says he will mount a legal challenge to the result of last week's presidential election, which he claims was rigged.