In Iron Man (2008), Tony Stark is captured in Afghanistan by a terrorist cell called the Ten Rings. The leader of the cell is a man named Raza. Many people were disappointed that the Ten Rings wasn't led by the Mandarin, but Raza is conspicuously shown fiddling with a large ring on multiple occasions, and the Mandarin has been known to lend his rings to underlings on certain occasions.
Two years later, Invincible Iron Man Annual 2010 goes through the Mandarin's history, which includes a retelling of Iron Man's origin. This is the first time the Mandarin has been tied to this story in the comics, and more importantly, Raza shows up as his right hand man, looking just like he does in the movie.
On the next page, we see his name is indeed Raza:
As a bonus, here's how Tony's escape is depicted:
Although the movie's Mark I armor is very similar to the Mark I armor in the comics (way closer than I expected it to look), it has a much more segmented, jury-rigged appearance as is depicted here. The armor in Iron Man's first appearance also didn't have flamethrowers. For comparison, here's the original Mark I armor from the comics and the relevant scene from the movie.
Tales of Suspense #39 (1963)
Iron Man (2008)