Nine minutes after departure, the capsule separated from its booster rocket and entered into its orbit.
The launch, from the barren steppes of Central Asia, took place under sunny skies at 0737 GMT. The craft is expected to reach the ISS on Monday.
Mr Tito, who is travelling with two Russian cosmonauts, is reported to have paid $20m to the Russians to become the world's first space tourist.
Live pictures from inside the craft showed the three men strapped into their seats during the final minutes before blast-off.
Feeling 'great'
Shortly after the launch, Mr Tito, wearing a white spacesuit decorated with an American flag on the shoulder was asked by a ground controller: "How do you feel Dennis?"
"Khorosho," he replied in Russian - "Great."
Nine minutes after departure, the capsule separated from its booster rocket and entered into its orbit.
Russian officials are already negotating terms for a second tourist to head for the ISS, the head of the Russian Space Agency has said.
Speaking at the launch centre in Baikonur, Yuri Koptev said negotiations were "under way," but refused to say who the would-be space traveller was.
"However he is unlikely to be a Russian," he said, in an implicit reference to the enormous fee paid by Mr Tito.
Good luck message
The launch was attended by the first female cosmonaut, Valentina Tereshkova, who went to the cosmodrome to wish the crew good luck.
She told them to smile back at all those who came to see them off and "remember that we are all waiting for you here down on land."
The US space agency, Nasa, had appealed for a delay because of computer problems on board the International Space Station which meant their Endeavour shuttle was still docked at the station doing repair work.
Problems fixed
Russia rejected any postponement of the take off, but finally agreed to delay Monday's scheduled docking if necessary.
With most of the problems fixed, Nasa says the new agreement "ensures continued safe operations".
Earlier, Nasa had warned that, if the Russian craft attempted to dock while the Endeavour was still in position, it could jeopardise the safety of everyone on board.
Under the new agreement, the Soyuz will maintain a safe distance from the ISS if the American space shuttle is still docked - but Nasa has expressed confidence that repairs will be completed before the Russian craft arrives on Monday.
Unwanted guest
The main task of the eight-day Soyuz mission is to replace the Soyuz escape craft docked to the ISS for a new one, as its service lifetime expires at the end of the month.
The dispute over the timing of the Russian mission follows a long-running disagreement over the inclusion of Mr Tito, a 60-year-old Californian businessman.
Mr Tito's $20m payment for his holiday in space is desperately needed by the Russians to maintain their space programme.
Nasa initially objected to Mr Tito's presence on the mission, saying he would be a safety risk.
Tough training
It was not until last Tuesday that the ISS consortium - which includes Russia, Canada, the US, Europe and Japan - granted an exemption to allow Mr Tito to participate.
Russia says Mr Tito, a former rocket engineer, has received the equivalent of a professional cosmonaut's training.
Mr Tito himself told reporters on Friday that "the training is tough, and it was made more difficult by political problems".
Before the eleventh-hour agreement with Nasa, Russia claimed computer problems had been blown out of proportion by US officials trying to put off Mr Tito's voyage.
Nasa's engineers have been working to correct the problems for several days.