Outside the medal places after a disappointing short programme on Wednesday in which Massot doubled a triple jump, the two performed a flawless free skate and fell to the ice in tears after finishing their routine.
After waiting anxiously for three more pairs to skate, they both wept when the gold was confirmed, with Savchenko, who claimed two Olympic bronzes with former partner Robin Szolkowy, sobbing in Massot's arms as he pressed his face into her hair.
Snapping off a flawless series of jumps and flowing choreography, the two broke their own world record in the free skate with 159.31 points for a total of 235.90.
Savchenko told reporters she had felt like the pair would be making history on Thursday and after their short programme, the duo regrouped and decided "we have nothing to lose and we will fight like tigers."
Silver went to Chinese world champions Sui Wenjing and Han Cong, who narrowly missed out with a total of 235.47, as Meagan Duhamel and Eric Radford secured bronze for Canada.
Duhamel, who with Radford won gold with Canada in the team event earlier in the Games, said that competition could have been the "secret weapon" that helped them to the podium in the individual event.
"We were comfortable in this arena, we were comfortable in this competition environment," Duhamel told reporters after what could have been her last competitive skate.
"We knew exactly what to expect. It felt like home. And that always makes you skate better and perform better."
The two-time world champions, who have said they would retire after the season, informed they are not planning to compete at the world championships next month.
After finishing second in the short programme, less than one point behind Sui and Han, Olympic Athletes from Russia Evgenia Tarasova and Vladimir Morozov were inconsistent in their free skate, with Tarasova falling on the throw triple Salchow.
Another inspiring performance was that of North Koreans Ryom Tae Ok and Kim Ju Sik.
Skating in front of a euphoric crowd, the pair were off sync on a double Axel but their inspired free skate still earned them 124.23 points, a personal best.
They finished 13th overall with a combined total of 193.63 points, the best score they have ever achieved.
(Reporting by Elaine Lies and Gabrielle Tetrault-Farber; Editing by John O'Brien and Sudipto Ganguly)
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