To have tests (and particularly integration tests) is a big help; as has already been mentioned.

Another thing is to use staged deployments: dont deploy to production-env from you development-env, but to a staging-env (if possible using the same or a copy of the live database). Then have a look with some (possibly non-technical) humans if the app still behaves as expected.

While it can be laborious for a large app, it should make a deployment to production a lot less scary.

As mentioned by others: upgrading is part of running a web app that depends on popular libs.