However, the astronauts on board the space station were forced to don protective gear after a harmful chemical started leaking from an oxygen vent.
Anousheh Ansari was accompanied by a US-Russian crew on the Soyuz TMA-9 capsule, which entered orbit about 10 minutes after lift-off from the Russian cosmodrome in Kazakhstan.
The American-Iranian reportedly paid $20 million (€15.76 million) to become the fourth private astronaut to take a trip on a Russian spacecraft and visit the station.
“I’m just so happy to be here,” she said.
Ms Ansari, 40, was due to return to Earth on September 29, along with cosmonaut Pavel Vinogradov and astronaut Jeffrey Williams, who have been on the station since April.
Following the alert on the station later in the day, NASA officials said the spill was not life-threatening.
“We don’t exactly know the nature of the spill...but the crew is doing well,” said Mike Suffredini, NASA’s space station programme manager.
The crew first reported smoke but it turned out to be an irritant, potassium hydroxide, leaking from an oxygen vent.
The crew donned surgical gloves and masks but did not have to put on gas or oxygen masks. NASA declared a spacecraft emergency for only the second time in the eight-year history of the station. The first time was for a false alarm of an ammonia spill.
The potassium hydroxide, a corrosive that can cause serious burns and can be harmful if inhaled, was cleaned up with towels and wrapped up in two rubber bags, Mr Suffredini said.
Ms Ansari follows in the footsteps of Briton Helen Sharman, who flew to Russia’s Mir Space Station in 1991 as a tourist as part of a lottery system called Project Juno.
Astronaut Lopez-Alegria said just a few years ago he was sceptical of private tourists. But he said it was clear that the Russian space programme needed such investment — without the Russian space programme, the US space programme would suffer.
Cosmonaut Mikhail Tyurin called Ms Ansari “very professional” and said he felt like they had worked together for a decade.