Neilson's team had made a habit of bouncing back throughout the tournament and did so again on Saturday as they responded to their 13-4 semi-final loss to Russia by beating China 7-3.
Even then they had to come from 3-0 down to do so, dragged along by their skip's best match of the competition.
Victory took Britain's medal tally in Sochi to six.
Neilson was in tears at the finish as she celebrated with her team-mates, Angie Malone, Jim Gault, Gregor Ewan and alternate Bob McPherson.
An emotional Neilson, originally from East Kilbride and now living in Strathaven, said: "That is the strength of this team. We can be down, some people might think we're out, but we are never down and out until the last stone is thrown.
"In our round robin we never lost two games back to back, so when we lost our game this morning that was our focus - we don't lose two games back to back.
"We came out and we fought hard and played the shots, and I am just delighted I was able to make some big shots when maybe through the week I hadn't."
Britain had lost 6-3 to the Chinese in their final round-robin match on Thursday, but in the bronze medal showdown at the Ice Cube Curling Centre, Neilson and her team rose to the occasion.
The skip was deadly accurate when it mattered at the end of a tournament in which her all-Scottish rink have been frustratingly inconsistent, mixing impressive wins with heavy defeats.
It was thanks to her they were not trailing by more than 3-0 after two ends, a rescue effort in the first limiting their opponents to a one-stone success.
A pin-point takeout from the 42-year-old in the third end made it 3-2 and in the next she forced an error from counterpart Wang Haitao to put her side 4-3 in front.
One-stone successes then followed in the fifth, sixth and seventh ends to leave China with too much ground to make up.
Neilson finished the match with an accuracy of 75 per cent.
Things had looked bleak after the semi-final loss, which ended in the team's third British record nine-stone defeat at these Games. But the team proved they had the character to respond.
"I always said the team that would be the most consistent over the week would be the one that would go home with a medal and I've maybe proved otherwise," Neilson said.
"We weren't the most consistent team but no matter what, we stuck together, through thick, through thin, through win, loss and it's shown."
Neilson also paid tribute to Frank Duffy, the skip of the Paralympic team which won silver in Turin in 2006. Duffy died in an apparent suicide in December 2010, killed in a car fire after being accused of alleged sex offences.
"Frank Duffy was a true inspiration," Neilson said. "When I watched the (Scotland) team in Glasgow (at the 2005 World Wheelchair Curling Championships) that was my inspiration and that medal's for him as well."
Malone, the only remaining member of the team from Turin, who was brought in for McPherson for the match, also justified her inclusion with 75 per cent accuracy.
She said: "This feels like a gold medal. This bronze medal feels like gold. We worked really hard this week, we dug deep. This is a gold medal for me."
At the Rosa Khutor Alpine Centre, sit-skier Mick Brennan, the former soldier who lost his legs in a suicide bomb attack in Iraq, rounded off his Games by taking 14th place in the giant slalom.
Ben Sneesby failed to finish after crashing on his second run, while, in the standing category, James Whitley placed 14th.