The $2.5 billion mission saw the one-ton, six-wheeled, nuclear-powered vehicle blaze through the pink Martian sky and touch down inside an ancient crater.
After a journey that had lasted eight months, and covered 352 million miles of space, Curiosity performed a series of aerial acrobatics before landing safely near the equator.
Jubilant scientists hugged, wept and distributed Mars chocolate bars to each other as one of them announced: "Touchdown confirmed. We are wheels down on Mars. Oh, my God."
Minutes later Curiosity sent back pictures showing one of it wheels, and its own shadow on the Martian surface.
Curiosity is Nasa's seventh landing on Earth's neighbour but the most ambitious and expensive.
Success had been far from certain and Nasa had labelled the entry, descent and landing the "Seven Minutes of Terror."
Two thirds of Mars missions to date had failed, including Britain's ill-fated Beagle 2 lander which was lost on Christmas Day 2003.
Adam Seltzner, leader of Curiosity's descent and landing team, had previously admitted to being "rationally confident but emotionally terrified."
Last night, he said he was "humbled" by the success of the mission, saying thank you "to the blue shirts".
"I think that this nation is a truly great representation of a piece of humanity that reaches out, explores and conquers," he added.
The landing was a pivotal moment for Nasa, which has faced cuts in its science budget and the controversial cancellation of its space shuttle programme. Officials had begun referring to it as the "Super Bowl of planetary exploration."
Following the successful landing President Barack Obama said it "marks an unprecedented feat of technology that will stand as a point of national pride far into the future,"
As it neared the atmosphere of Mars the Mini Cooper-sized spacecraft accelerated with the pull of gravity and made a fiery entry at a speed of 13,200mph, 17 times the speed of sound.
It was then slowed down with the help of a giant supersonic parachute. A hovering, jet-powered "sky crane," then descended towards the surface, lowering Curiosity to the ground on three 25ft nylon tethers.
It landed upright on all six wheels and the sky crane cut the cords before powering away and crashing at a safe distance.
Nasa Administrator Charles Bolden said: "It's incredible, it doesn't get any better than this. I was a basket case. I was on pins and needles. It's a huge day for the nation and the American people. It belongs to all of us."
Clara Ma, the schoolgirl who named and signed Curiosity before it was blasted off into space, said: "That was so special to me. I was crying, laughing. I'm so happy and just getting to see the reactions of the mission team members hugging each other, crying, laughing. I got really emotional."
Curiosity will search for evidence that Mars may once have held the necessary building blocks for microbial life to evolve.
It has an array of sophisticated chemistry and geology gadgets for analysing soil and rocks.
Those include a laser gun that can hit a rock from 23ft, creating a spark. The spectral image from the spark is then analysed by a special telescope to establish its chemical composition.
It also has cameras and a robotic arm with a power drill, and a magnifying imager that can reveal details smaller than the width of a human hair. Samples will be analysed using a state-of-the-art on-board laboratory.
Curiosity will face several weeks of health checkups before taking its first short drive on the Martian surface.
Dr Charles Elachi, director of the Jet Propulsion Laboratory, said: "Tomorrow we will go out and start exploring Mars, next week and next month and next year we will be bringing new discovery [sic] every day every week.
"We are not only going to continue exploring Mars but exploring the solar system and exploring the universe. Because our curiosity has no limit."
Its landing site in Gale Crater was picked because it may once have been a large lake. The crater includes a three-mile high mountain, Mount Sharp, the base of which appears rich in minerals that formed in the presence of water. Curiosity will look there for basic ingredients essential for life, including carbon, nitrogen, phosphorous, sulphur and oxygen.