In his inaugural speech, Mugabe said: "Once again we have shamed all our detractors."
Movement for Democratic Change (MDC) leader Tsvangirai rejected Mugabe's invitation to attend the swearing-in and dismissed it as meaningless.
"I can't give support to an exercise I'm totally opposed to . the whole world has condemned it, the Zimbabwean people will not give this exercise legitimacy and support," he said.
Tsvangirai added he would ask African Union (AU) leaders meeting in Egypt on Monday not to recognise the re-election.
His party urged the AU not to welcome Mugabe at the summit, and MDC vice president Thokozani Khupe said she was not holding talks with the government delegation at the meeting.
Mugabe says he will confront his African critics at the meeting, which he will attend with his new mandate.
Mugabe is under pressure from within Africa to enter talks with Tsvangirai to end his country's deep crisis, which has ruined a once-prosperous economy with hyperinflation and sent millions of refugees fleeing into neighbouring states.
In an apparent response to that pressure, he said in his inaugural speech that he was committed to dialogue with the MDC.
Tsvangirai said the opposition was also committed to AU-sponsored talks, although no negotiations had started.
The AU seems reluctant to back Western calls for sanctions, favouring instead a Kenya-style power-sharing transition.
The Egypt summit may be split between critics of Mugabe, like Kenya, and opponents of any action against him led by South African President Thabo Mbeki, who has been widely criticised for taking a soft line with his neighbour.
Mugabe paid tribute to Mbeki, describing him as a statesman. Mbeki has tried to mediate an end to the crisis since last year.
South Africa's trade union confederation COSATU, a fierce critic of Mugabe and Mbeki, said in a statement that the AU must refuse to recognise the Zimbabwean leader, who had won a "farcical" election by intimidation, violence and murder.
Kenya's Prime Minister Raila Odinga was quoted as saying on Sunday that the AU should deploy troops but AU security chief Ramtane Lamamra played down the prospects of any peacekeepers.
Alister Sparks, a political analyst at Standard Bank, said Mbeki had lost credibility as a mediator and Mugabe's victory would send a new influx of refugees into neighbouring countries.
"I would hope that no civilised country in the world would recognise it (the election)," he told Reuters.
