Andrew Weibrecht, starting 29th with all the favourites gone, used a pair of U.S. team mate Miller's skis to go one better than his third-place finish in Vancouver and win a surprise silver.
Canada's Jan Hudec finished with the exact same time as Miller, one minute 18.67 seconds, in the second Alpine race of the Games to see a tie for a medal. The women's downhill had two golds awarded on Wednesday.
"At the start I told myself that I have had a nice Olympics so far but that more was still possible," said Jansrud, who won a bronze in last Sunday's downhill.
If Norwegian gold had looked assured after the first 25 had skied, Weibrecht produced a late scare with Miller giving useful feedback about the course.
"When Andrew came down he scared me," laughed the Norwegian. "It was a little too exciting. My legs were like jelly there for a second but I knew I had finished very strong."
Jansrud's triumph was Norway's fourth in a row in the discipline dating back to 2002 - equalling Austria's men's record of four successive slalom titles between 1952 and 1964.
Now-retired Kjetil Andre Aamodt began the sequence in Salt Lake City and repeated the feat in Turin in 2006 to become the oldest Alpine Olympic champion and medallist at the age of 34 years and 169 days.
Aksel Lund Svindal won in 2010 but was only seventh on Sunday in a major disappointment for a man who arrived as a likely multiple medallist but has so far failed to get on the podium.
"It could have been double if I didn't mess up the first three gates," said Svindal.
"But I messed up and was not good enough and lost a medal on the first three gates. I knew after my run Kjetil could go a lot faster and I'm glad he did."
World champion Ted Ligety of the United States also failed to live up to his advance billing, finishing 14th.
Miller's bronze was the sixth Olympic medal of his career, in his fifth Games, and he secured it in typical fashion - by the skin of his teeth after risking all on his way down the gleaming Rosa Khutor slope.
The American was 13th out of the start hut and it looked like gold could be a possibility, despite a big mistake at the last jump, until Jansrud came down as the 21st starter and pushed him off the top spot.
Hudec, only four years younger than Miller, immediately followed Jansrud down before Weibrecht added a late twist.
There were emotional scenes afterwards for both Americans, with Miller breaking down in tears and comforted by his wife at the finish.
He explained later that he had been thinking of his snowboarding younger brother Chelone, who died of an apparent seizure last year.
"Losing my brother this last year was really hard for myself, my family," he said. "It was just a lot of emotion. To have things go well today, as well as they did...everything felt pretty raw and pretty connected."
Weibrecht, nicknamed both 'Wombat' and 'Warhorse' by his team mates, said Sunday had been "one of the most incredible moments" of his life, coming after multiple surgery and a difficult season in the World Cup.
"This is probably the most emotional day of ski racing I have ever had," he said, having cursed his late start number after the draw on Friday night as a tough challenge.
"There have been times when I have had to evaluate whether this is really what I want to do. Even as recently as yesterday."
Hudec was the first Canadian man to medal in Alpine skiing at the Olympics since Ed Podivinsky's bronze in downhill in 1994.
- Sports & Recreation
- Andrew Weibrecht