Hincapie trails 5 seconds behind overall leader Rinaldo Nocentini of Italy, who was 5 minutes, 36 seconds behind Ivanov but kept the yellow jersey for an eighth straight day.
Alberto Contador, the 2007 winner, dropped from second to third overall, 6 seconds behind Nocentini. Armstrong is 8 seconds back of the leader.
Hincapie is the only rider to be a teammate of Armstrong as he won seven straight Tour titles from 1999 to 2005.
"George is a good friend, We wanted him to have the jersey -- it would have been great for him," Armstrong said.
Hincapie has previously donned the race leader's shirt: He once wore it for a day in 2006 -- the year after Armstrong retired.
A police motorcycle struck a woman crossing the course route at the 38-kilometer mark (24-mile) during the 199-kilometer (124-mile) stage from Colmar to Besancon.
After hitting the woman, the motorcycle skidded into two other spectators, Tour organizers said. A 36-year-old complained of neck pain, and a 61-year-old broke a leg. They were taken to hospital.
The 36-year-old Hincapie, who rides for the Columbia team, entered the stage 5:25 behind Nocentini in 28th place. Through the last kilometers, Hincapie's breakaway group was more than 6 minutes ahead of Nocentini, fanning suspense about the yellow shirt's future.
Nocentini's AG2R-La Mondiale team, then American team Garmin both accelerated the pace, trimming the gap by the finish line -- so that Hincapie missed out the yellow jersey by just seconds.
French TV cameras showed Hincapie as he watched Nocentini's pack cross the finish line -- and he was frustrated at not taking the yellow shirt. Hincapie didn't speak to reporters before entering the team bus.
Armstrong, on his Twitter feed, wrote: "No one wanted George in yellow more than me."
"Until 10km to go he was solidly in yellow until GARMIN put on the gas and made sure it didn't happen," Armstrong wrote. "(Hincapie) deserves to be in yellow tonight. He deserves more than that."
For Ivanov, the day was much more fulfilling. He made a sign of the cross as he finished the stage in 4 hours, 37 minutes, 46 seconds after escaping the breakaway group with 11 kilometers left.
Ireland's Nicolas Roche, son of former Tour champion Stephen Roche, was second, and Hayden Roulston of New Zealand was third. They crossed along with Hincapie, 16 seconds back of Ivanov.
The 34-year-old Russian national champion, who also won a Tour stage in 2001, collapsed on to a curb after finishing so he could catch his breath.
"It's an amazing victory," he said. "I didn't have any more air. I was just two minutes, breathing, breathing ... I was very tired.
"I gave all my energy on the last 10 ks."
Ivanov rose to 56th place overall, 27:47 behind Nocentini. The Russian had entered the day's stage in 62th place, 33:23 back.
The race moves into the Alps for Sunday's 15th stage, a 207.5-kilometer (128.9-mile) ride from Pontarlier, France, to the Swiss ski station of Verbier.
The Tour ends July 26 in Paris.