Duels fought in the early American West involved the duelers facing each other with guns in their holsters, and hands poised over the guns. Each dueler would try to remove ["draw"] his gun and fire faster than the other. So, to "beat to the draw" was a very important objective. So, yes; Jake got the job because he was quicker, but at least you're alive to tell about it.

it means to do something before someone else does it cowboy...

Hi you can also say to beat someone to the punch as well as to beat someone to the draw It means to do something before someone else does it. I wanted to have the new car, but Sally beat me to the punch. I planned to write a book about using the new software program, but someone else beat me to the draw. In your example Jake got the job before you.