Oudin replaced Kalu in the 14th minute after the 23-year-old Nigerian fell to the ground on a hot evening at the Stade Velodrome, with medics rushing on to the pitch with a stretcher to attend him.
Having been shielded by his team mates after he collapsed, Kalu was able to get up and walk off the pitch with their help while holding an ice pack against his head.
The striker, who also collapsed from severe dehydration during a Nigeria training session at the 2019 African Cup of Nations, resumed playing several moments later but appeared groggy and was eventually substituted.
"According to the doctor, it was just a dizzy spell," said Oudin, who netted Bordeaux's equaliser as they fought back from two goals down to earn a 2-2 draw.
"He is to take more tests and we will know more tomorrow [Monday] during the day. It gave us a bit of a chill and it's still quite shocking.
"We inevitably thought about what happened with [Denmark midfielder Christian Eriksen], the whole team and everyone was scared. We're happy for Kalu, everything is fine and that's the most important thing."
The stadium went silent during the incident, which brought back memories of Eriksen's collapse at Euro 2020 in June when he suffered a heart attack in the first half of Denmark's match against Finland in Copenhagen.
Eriksen's life was saved when CPR was administered to him on the pitch and his heart was re-started with a defibrillator before he was taken to hospital, where he recovered after undergoing heart surgery.