In hospitals, alarms on patient-monitoring devices create a cacophony of noise day and night—beeping, pinging and ringing so often that doctors and nurses ignore them, turn them off or just stop hearing them.

Now, hospitals are adopting solutions to silence or eliminate unnecessary alarms, while ensuring that staffers don’t miss alerts that could signal a life-threatening crisis. Smarter technology and more-precise monitoring practices are helping prevent false alarms, alert nurses to true emergencies, and identify deteriorating...