The electoral commission has not yet responded to Mr Odinga's accusation, but politicians have called for calm. Many fear a repeat of the violence after a disputed election 10 years ago.
More than 1,100 Kenyans died and 600,000 were displaced following the 2007 vote.
Electoral officials say that with 91% of results in, Mr Kenyatta is leading with about 54.5%, to Mr Odinga's 44.6%.
These results mean Mr Kenyatta appears to be heading for a first-round victory.
In order to avoid a run-off, a candidate needs 50% plus one of the votes cast and at least a 25% share of the vote in 24 of Kenya's 47 counties. There were eight candidates in all, but apart from Mr Kenyatta and Mr Odinga, none polled more than 0.3% of the vote.
Speaking at a press conference Mr Odinga, who heads the National Super Alliance (Nasa), said his supporters should remain calm.
But in the opposition stronghold of Kisumu, in western Kenya, a group of several hundred of his supporters took to the streets shouting: "No Raila, no peace".
They were then dispersed by police using tear gas.
Riot policemen have been seen on the streets of Kisumu, western Kenya
What is Mr Odinga's complaint about the vote?
Mr Odinga said that the hackers had gained access to the Independent Electoral and Boundaries Commission (IEBC) computer system by using the identity of the commission's IT manager, Chris Msando, who was killed last month.
He alleged that the hackers "loaded an algorithm" that allowed them to alter the results and ignore the figures being transmitted from tallying centres around the country.
Election officials have been busy through the night counting the votes
The opposition leader had earlier told journalists the results coming in were "fake", because the authorities had failed to present documents verifying the results.
What does the electoral commission say?
The electoral commission has not reacted to the claims that its IT systems had been hacked, but earlier it urged people to wait calmly for the full results of Tuesday's vote.
"During this critical phase, we urge all Kenyans to exercise restraint as we await official results from the polling stations and indeed as they start trickling in," the commission said.
Media captionInside a Nairobi polling station with the BBC's Anne Soy
However, it admitted that a lack of mobile data coverage had delayed the delivery of the supporting documents, which had formed the basis for the opposition's earlier complaint.
There had been reports on election day of the failure of some voter-identification equipment.
Also, one in four polling stations were without strong mobile phone coverage, meaning that officials were asked to drive to the nearest town to send results.
The presidential candidates' agents would have "special access" to the forms, though, the electoral commission said.
The Daily Nation newspaper quoted commission head Ezra Chiloba as saying only results supported by the forms had been published.
How did the voting go?
Voting passed off largely peacefully and some polling stations remained open after the scheduled 17:00 (14:00 GMT) closing time.
People started queuing early to ensure they could cast their vote. Long queues could be seen, and video footage at one polling station showed people injured after an apparent stampede.
People began queuing early in the morning and even overnight to cast their votes
There were reports that one man had been killed in clashes in the Kilifi area.
But there was one heartening moment when a woman gave birth to a baby girl as she queued in West Pokot to cast her ballot. New mother Pauline Chemanang called the circumstances of the birth a "blessing" and called her baby Kura, Swahili for "ballot", according to local radio.
However, in the run-up to election day, there were claims of vote-rigging and hate speech flyers and rhetorical text messages began circulating.
Some nervous Kenyans stockpiled food and water, while police prepared emergency first aid kits in the event of violence.
What is at stake?
Mr Kenyatta is hoping for a second term in office.
Voting for the national and local assemblies has also been taking place.
Mr Kenyatta called on Kenyans to pull the country together
Mr Odinga, 72, has run for president three times and lost each time. President Kenyatta, the 55-year-old son of Kenya's founding president, narrowly beat him in the last election in 2013, but their rivalry is generations old - their fathers were political opponents in the 1960s.
Mr Kenyatta and his running-mate William Ruto were indicted by the International Criminal Court for their alleged roles in the bloodshed a decade ago. The case ultimately collapsed due to lack of evidence, and after key witnesses died or disappeared.
Kenya's election in numbers:
- Six separate ballot papers: For president, national assembly, female representatives, governors, senate and county assemblies
- 47 parliamentary seats and 16 senate seats reserved for women
- Eight presidential candidates: President Uhuru Kenyatta and opposition leader Raila Odinga are favourites
- Kenyatta beat Odinga in 2013 - their fathers were also political rivals in the 1960s
- A candidate needs 50% plus one vote for first-round victory
- More than 14,000 candidates running across the six elections
- More than 45% of registered voters under 35
- Some 180,000 security officers on duty nationwide in case of trouble