No...one team starts at their opponents 25 yard line. They score as many points as they can. After they score (or not) the other team starts at the 25.
For example....first team gets close enough to kick a field goal...they lead by 3.
The opponent knows if they get a TD on their poession, they win.
If all they can do is kick a goal as well, you go to 2nd overtime.
Repeat as necessary.