A potentially damaging 13-4 loss to hosts Russia in the semi-final had skip Neilson deflated, but her team responded in style, earning bronze with a 7-3 victory against China in a match in which they had to come from behind.
A slow start saw the British rink go 3-0 down after two ends, but seven unanswered points from the next five marked an emphatic response for bronze at the Ice Cube Curling Centre.
The Sochi 2014 medal is Britain's first in wheelchair curling at the Games since silver at Turin 2006 and, finally with something to show for their hard work, Neilson, from East Kilbride, believes her rink's willingness to never give up was their biggest strength.
"It has been an amazing experience and we are delighted to be coming back with a medal," she said. "I always said the teams that were the most consistent over the week would be the ones that would go home with medals.
"But maybe I have proved otherwise because we weren't the most consistent team, but we had enough wins to get us through to the play-offs. We stuck with our plan and our team dynamics, you will have seen that on and off the ice.
"No matter what, we stick together, through thick and thin, through win and loss, and it has shown. We showed the strength of this team. We can be down and some people might think we're out, but we are never down and out until the last stone is thrown.
"In the round-robin stage we never lost two games back-to-back, so when we lost our game in the 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 that I was able to make some big shots when maybe through the week I hadn't."
Looking to guarantee themselves a medal at the Ice Cube Curling Centre with victory over the Russians, Britain fell 4-0 behind after just two ends and were always playing catch up thereafter.
A magnificent shot by Neilson earned them a three in the third end to get the British rink - the other members are fellow Scots Jim Gault, Angie Malone, Bob McPherson and Gregor Ewan - right back into the match.
However, a disastrous fourth end followed with Britain conceding a record seven stones to effectively end their hopes of reaching the final.
China were beaten by Canada in their own semi-final, with Jim Armstrong's rink defeating the hosts Russia in the final to take their third gold from as many Winter Paralympic appearances.
Despite the Chinese taking a 3-0 lead, the British were not deterred and fought back with a crucial two in the third end and after that a succession of steals, thanks to some very important shot-making from Neilson.
"We went in with a game plan and we stuck to that game plan. We tried it in the morning and it didn't quite work, but we stuck to it," Neilson said.
"I have a key ring on my bag and it says 'wish it, dream it, do it'. I wished it, I dreamt about it and now we've done it and we've got the medal to take home. I am quite an emotional person and I have managed to hold it together until now, but I don't care, I am happy to let my emotions show."
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.
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