Follow Dom as he knows the stealthy route. If spotted you can restart the checkpoint and still get the achievement.
When you get to the bridge with the Brumak, go right and through the cave to remain unseen. You'll then go under the bridge.
As long as you stay undetected until you rescue the COG prisoner, you'll get the achievement. It goes full on from there - you wouldn't be able to go stealthily if you wanted too!
The achievement pops right at the end of the level on the title screen but you have to go through a lengthy firefight first.