The 24-year-old Serb seemed to be heading for a routine victory when serving at 5-4 in the opener but then the game turned on its head.
The teenage American wasted three set points during a monumental tie-break, and despite losing it 10-8, Jankovic's game slowly crumbled with her movement increasingly restricted.
An injury and medical time-out lasting nearly 12 minutes suggested she had problems and Oudin took full advantage with her ferocious forehand to win a nervy second set.
Although the 2008 US Open runner-up exhibited occasional glimpses of her power-game, the lively American, the youngest player left in the draw, never stopped chasing.
After amazingly dominating Jankovic from the baseline, she flew through the third set to secure a fully-deserved victory to set up a clash with Polish 11th seed Agnieszka Radwanska.
A thrilled Oudin said: "I think I handled it really well. I was just thinking she was any other player and this was any other match at any other tournament, not like on the biggest stage at Wimbledon playing my first top 10 player.
Not the greatest of birthday presents for Kuznetsova on Court One
"I knew I was right there with her every single point, so I knew I could do it if I just kept trying and kept fighting.
"I'm very excited right now. But I'm hoping there will be better days too."
Jankovic said she was not affected by the heat but could not recover from feeling ill during the match.
"The conditions were tough but that was not my problem," she said. "In the first set I felt really dizzy and felt like I was going to end up in the hospital. I started to shake and that I was losing my consciousness.
"I thought I was going to fall down or have to call the ambulance or leave the court but I came back and started to fell a little bit better.
"However I felt weak and did not have the power in the shots and I was not the same player.
"It was very difficult for me to play in the circumstances and it made me as well quite nervous. I tried my best but did not have enough energy to hit the shots and go after the shots."
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