The US skier can take the outright record on Saturday if she triumphs in a slalom in the Swedish resort where she took her first victory in 2012.
Whereas Stenmark's success in the 1970s and 80s was limited to slalom and giant slalom, Shiffrin has also won in downhill, super-G, combined and parallel as well as the two technical disciplines.
"This is just a spectacular day," she said after celebrating at the finish. "Oh my goodness."
Shiffrin, who turns 28 on Monday, has already secured the overall World Cup title for a fifth time as well as the slalom crystal globe.
She was more than half a second clear of Canadian Valerie Grenier (26) after the first leg and stretched her advantage in the second to beat Italy's Federica Brignone (32), who was fastest in that run, by 0.64.
Sweden's Sara Hector (30) was third, 0.92 slower than Shiffrin's combined time of one minute 54.64 seconds.
The 100 points for a win gave Shiffrin an unassailable lead of 214 points in the giant slalom standings with only one race remaining at next week's World Cup finals in Andorra.
Shiffrin has 1,928 points in the overall standings with Petra Vlhova (27) of Slovakia second on 1,025.
She knew the giant slalom globe was hers when Switzerland's Lara Gut-Behrami (31), the closest rival before the race, failed to finish the first run and Vlhova was unable to score enough with her second effort.
"I knew the GS globe was locked in before I raced so then I just wanted to push and kind of fight for it," said Shiffrin, who has now won six out of nine giant slaloms this season as well as world championship gold.
"It was harder with the visibility. For the first run I had luck with the light so the second run was much more dark and I just kept trying to push."
The win was the 20th of her career in giant slalom, equalling the women's record in the discipline of retired Swiss skier Vreni Schneider.
Although she said the GS globe had been her biggest dream for the rest of the season, there was no doubting the importance of the landmark win as Shiffrin stood on the podium holding a pair of skis with the number 86 on the base.
"I think now we will send some messages back and forth," she said of Stenmark, before addressing the absent Swede via a television interview.
"No matter what I do, it doesn’t compare to what you achieved," she said.
"Maybe I get the 87th victory, maybe not, but for me the biggest dream is to be mentioned in the same sentence as you. It’s pretty special -- who you are, who you were as a ski racer, what you achieved as a human.
"That’s the most inspiring thing. You may not want attention, but you still get it."
Shiffrin had taken her 85th win on January 28 in the Czech resort of Spindleruv Mlyn after breaking compatriot Lindsey Vonn's women's record of 82 earlier that month.
