Giglia won gold in the C1-3 3,000m cycling race on the opening day of the games, beating American Jamie Whitmore into second place.
She said she had taken "extra motivation" from 10-year-old Alasdair Rowan, who suffered a stroke in July and is now recovering.
At the time of the last Paralympics in London, Giglia was a sports instructor.
But in January 2013, four months after the games had ended, she had a stroke and brain haemorrhage, leaving her with restricted movement down her right side.
Cycling formed part of her recovery and she told the BBC in August: "I never dreamed I would get to this level, but it has allowed me to turn what was a bad situation into a good one.
"It gives me something to live for and I hope what I am doing can give other people hope that they can do something with their lives."
Giglia, who lived in Stratford-upon-Avon before moving to Manchester where British Cycling is based, dedicated her Paralympics victory to other stroke patients trying to come to terms with the effects and, in particular, to one 10-year-old schoolboy who had inspired her.
Writing on her Instagram page before the race, she said: "My Paralympic Games kicks off tomorrow and every race I compete in I'm dedicating to individuals and their families who are all currently dealing with the after effects of a stroke!!!
"Tomorrow is the 3km pursuit and is dedicated to a strong and determined young man going by the name of ALASDAIR ROWAN who is on his road to recovery and given me extra motivation to succeed...This one's for you Alasdair."
Alasdair's father, Lee, told the BBC that Giglia had asked his wife, Fiona, if she could dedicate the race to their son after they were put in touch by charity Different Strokes.
Giglia is a patron of the charity.
Alasdair, then aged nine, had a stroke in July after suffering an aneurism.
He has since returned home following an emergency operation and a month-long stay in hospital, and is hoping to return to school soon.
Mr Rowan said the family had been cheering Giglia on to victory and had been filled with "nervous excitement" when they saw her carrying a picture of Alasdair as she was about to be interviewed.
Mr Rowan, from Bulkington, in Warwickshire, said he was pleased that she had used the interview to raise awareness about young stroke patients.
"A lot of people think strokes only happen to older people - that certainly is not the case. Alasdair is younger than normal but he's not unique."
Giglia began her cycling career after she was spotted at a British Cycling talent identification event and joined its development programme in May 2014.
She had her first major success in international competition at the Newport Para-cycling International in early 2015, when she won silver in the individual pursuit and bronze in the 500-metre time trial.
Following this performance, she was selected for her debut UCI Para-cycling Track World Championships in Apeldoorn, Netherlands, and finished fourth in the time trial and individual pursuit.
She then achieved her first podium finish in road cycling by coming third in both the time trial and road race at the UCI Para-cycling Road World Cup in Maniago, Italy, in June 2015.
Two fourth places in the same events at the UCI Para-cycling Road World Championships followed later that summer.
She finished the year by winning a gold in the C3 pursuit race at the Manchester Para-Cycling event.
In March this year she won two titles at the UCI Para-cycling World Championships in Italy, winning both the C3 500m time trial and the C3 3km pursuit in world record times.
Speaking to Channel 4 after her Paralympic triumph, Giglia said: "It hasn't sunk in yet. I wouldn't be here without my team-mates, and my back-up team behind me.
"I didn't think I would make it to Rio and I thought it was a bit ambitious, but I wanted to give it a go."
Megan Giglia, who won Britain's first gold medal at the Rio Paralympics, says sport gave her something to live for after she suffered a stroke and a brain haemorrhage aged 27.