Hassan battled with Tigst Assefa of Ethiopia and managed to shake her off in the last stretch, crossing the finish line in an Olympic record time of two hours, 22 minutes and 55 seconds.
Assefa, three seconds behind her, took silver and Kenya's Hellen Obiri claimed the bronze in 2:23:10.
Hassan's victory capped a gruelling Olympics schedule as she became the first athlete to medal in all three distance events at a single Olympics since Czech Emil Zatopek in 1952.
"I feel like I am dreaming," Hassan said. "At the end I thought, 'This is just a 100m sprint. Come on, Sifan. One more. Just feel it, like someone who sprints 200m'."
Hassan had initially signed up for the Olympic 1,500 metres as well before deciding that three events would be enough of a challenge.
Earlier this week Hassan said that finishing the marathon is "a kind of hell", and on Sunday she collapsed after the finish line before picking herself up, beaming and waving as the crowd cheered.
"Every step I challenged myself, and now I am so grateful I didn't push myself too much on the track. I was scared of this race," she said afterwards.
The race had been impossible to call with all the favourites among a leading pack up until the last 10km.
Defending champion Kenya's Peres Jepchirchir then dropped back as Assefa ploughed ahead with a handful of challengers with Hassan hanging onto the back of the pack in much the same way she does on the track, biding her time before pouncing.
Sprint Finale
She delivered her devastating turn of pace on the last turn, sneaking past Assefa on the inside and briefly knocking elbows with the Ethiopian, who tried to rally but could not catch up.
Ethiopia filed a request to disqualify Hassan for obstruction after she collided with Assefa in the final sprint, but the request was rejected.
Assefa told a news conference Hassan had pushed her but congratulated her on the win, speaking through an Amharic translator. Asked if the collision had ruined her shot at gold, she said "Yes". Hassan did not attend the news conference.
Assefa, 27, had looked likely to make it an Ethiopian sweep of the Olympic marathons after Tamirat Tola won the men's race on Saturday, as she surged up the course's steep hills looking calm and collected throughout.
Assefa has two of the fastest-ever marathon times and took more than two minutes off the world record in a mixed race to win last year's Berlin marathon.
She had to settle for silver on Sunday, though, in a repeat of her second-place finish at the London marathon in April when Jepchirchir prevailed.
Obiri, 34, ran a masterful race to clinch bronze, powering on despite having to stop and throw up at one point.
"I felt so bad in my stomach. I do not know - maybe because of that, I stopped and puked," she said. "Then, to catch up with the girls, I used a lot of energy."
Like Hassan, Obiri has added the 42km to her scintillating middle-distance career, having won silver in the 5,000m in Rio and Tokyo. She has won the Boston marathon twice and New York once.
Tackling the exceptionally hilly Paris course on a hot morning, the athletes grabbed bags of ice at each pitstop to cool themselves off.
Hassan's last-minute victory was a fitting climax to the Olympics' athletics programme, which organisers switched up to finish on the women's marathon instead of the men's, which was run on Saturday.
The Paris course, taking athletes out to the Chateau of Versailles and back, was designed to commemorate a protest during the French Revolution in which thousands of women marched on Versailles to demand bread from the king.
