General:
* Live through
* Endure
* Persist
* Last
* Pull through
* Make it
* Come through
* Weather
More specific:
* Outlive (emphasizing living longer than something else)
* Overcome (emphasizing conquering a challenge)
* Prevail (emphasizing success in a difficult situation)
* Escape (emphasizing avoiding danger)
* Withstand (emphasizing resisting something harmful)
The best word to use will depend on the specific situation and the meaning you want to convey.