How do anesthesiologists keep you alive while you are under anesthesia?

Advances. Modern monitoring allows anesthesiologist to analyze your blood pressure, pulse, ekg, oxygenation level and many other values if nessasary. It makes modern anesthesia very safe. Advances in pharmacology allow use of safer medications and airway management relies on expertise of many generations of anesthesiologists.
Vigilance. Once you are under anesthesia your doctor will take over your breathing and temporarily place you on a ventilator. We watch all your vital signs throughout the case and are ready to intervene to treat blood pressure changes, heart rate changes and the ability to provide oxygen to the patient. We also keep an eye on what the surgeon is doing and its affects on you. Hope this helps.
Anesthesia. We keep you alive by protecting your airway and respirations; by keeping your heart beat and blood pressure within acceptable levels. All these we do by titrating medications.
Use your body. Under general anesthesia, your muscles may be paralyzed and your brain is asleep or has amnesia. The anesthesiologist helps maintain your airway so you can get proper oxygen. They also watch your blood pressure to be sure your heart function is not affected negatively, as your heart continues to support you. Your airway is protected so nothing can aspirate into your lungs.