Wiktionary
sigh of relief
n. 1 A release of stress through breathing motions. 2 (context idiomatic English) A reassurance or support, something that reduces stress from an arduous activity.
Usage examples of "sigh of relief".
Connor reached for the phone, -and Laura let out a sigh of relief.
He breathed a sigh of relief: a gust of wind had come through the broken window and pushed the door shut.
Her gaze on Hanna had such a benign aspect that Hanna gave a sigh of relief, though it hurt to let air whistle from lungs to mouth.