Lyles, who had been bidding to complete a 100m-200m double in Paris, was taken off the track in a wheelchair following the race, won by Botswana's Letsile Tebogo.
The 27-year-old Lyles, who has suffered from asthma in the past, had been seen wearing a surgical mask shortly before Thursday's 200m final.
However, there had been no sign of illness when the American was introduced to the crowd before the race, bounding onto the track and gesticulating with his arms to pump up the crowd.
In an interview with NBC, Lyles said: "I woke up early, about 5:00am on Tuesday morning and I was feeling really horrible.
"I knew it was more than just being sore from the 100 (metres). We woke up the doctors and we tested and unfortunately, it came up that I was positive for Covid."
Lyles later told reporters that his illness "definitely affected my performance".
"But to be honest, I'm more proud of myself than anything," he added. "Coming out and getting the bronze medal with Covid.
"We tried to keep this as close to the chest. Only people who knew were the medical staff, my coach, my mom, my family. We didn't want everybody to go into a panic.
"We wanted them to be able to compete. And you don't really want to tell your competitors you're sick. Why would you give an edge over you?"
USA Track and Field said in a statement that US team officials had "enacted all necessary protocols to prioritise his health, the wellbeing of our team, and the safety of fellow competitors" following Lyles' positive test on Tuesday.
"Our primary commitment is to ensure the safety of Team USA athletes while upholding their right to compete," the statement said.
"After a thorough medical evaluation, Noah chose to compete tonight. We respect his decision and will continue to monitor his condition closely.
"As an organization, we are rigorously adhering to... guidelines for respiratory illnesses to prevent the spread of illness among team members, safeguarding their health and performance."
