Just because Bashir shot the Klingon doesn't mean that it saved Odo's life. It could have just been the writers opting to give Bashir a moment of heroism in the fight, since he was the only character who hadn't already had a moment in the fight to really show off his fighting skill, because he's a doctor, not a warrior. But everyone else had one (well, except Quark and Jake, but they were civilians who weren't involved in the fight, so...), so they threw in a little moment for Bashir.
The episode unambiguously plays it up as Bashir saving Odo's life. They had a conversation about Odo watching his back in an earlier scene, the attack was clearly portrayed as Bashir shooting a Klingon about to kill Odo, and the scene is finished off with a grateful Odo thanking Bashir. There is absolutely no other way to read the scene.