A final deal on power-sharing was reached in January, after Mr Tsvangirai returned to Zimbabwe following an absence of more than two months for fresh talks with Mr Mugabe.
The cabinet in the new coalition government will be sworn in on Friday.
There is deep scepticism about whether it will work, says the BBC's southern Africa correspondent, Peter Biles, in Johannesburg.
At best it will be a transitional arrangement leading eventually to a new constitution and fresh elections, he says.
Zimbabweans also reacted with cautious optimism.
Harare businessman Ian Stephens said it was too early to celebrate.
"It depends on how co-operative Mugabe is and whether he can be trusted. But he no longer has absolute power and that could be the turning point," he said.
UK Foreign Secretary David Miliband welcomed Mr Tsvangirai's inauguration as a "step forward", but voiced concern that Mr Mugabe remained as president.
"While Morgan Tsvangirai is acknowledging the crowds, behind him is a lurking figure and that figure is President Mugabe, who has tyrannised that country and bought it to its knees," he said.
The international community stood ready to offer additional aid to Zimbabwe but it depended on the actions of the new government, he said.
A US State Department spokesman congratulated Mr Tsvangirai but said sanctions would stay until President Mugabe showed he was sharing power.
South African President Kgalema Motlanthe said Mr Tsvangirai's swearing in "is a vindication that our approach to the crisis of Zimbabwe all along has been correct, despite scepticism in certain quarters".
Mr Motlanthe called on the international community to lift sanctions on Zimbabwe and turn its attention to Zimbabwe's humanitarian crisis.
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