You should understand how anesthesia is typically given. The typical sequence goes like this. The patient first receives a short-acting intravenous anesthetic drug (these days typically propofol), which lasts for 3-6 minutes at most, and which puts the patient to sleep. While the patient is asleep from this, the anesthesiologist gives oxygen, followed by a drug that paralyzes the muscles (typically pancuronium or vecuronium, but there are many). As the still-asleep patient stops breathing from the paralytic drug, the anesthesiologist places a breathing tube (called an endotracheal tube) in the patient’s windpipe through the mouth and starts to breath for the patient with a machine attached to the tube. Next, while the patient is still asleep, the anesthesiologist turns on an anesthetic gas that comes through the machine and goes into the breathing tube. It is the gas that keeps the patient asleep from now on. The anesthesiologist then gives additional paralytic drugs as needed, plus usually a narcotic pain-killer, typically fentanyl. When it is time for the patient to wake up, the gas is turned off, and the patient awakens within minutes.
I describe all of this because your scenario would be a difficult one to pull off in real life. Giving anesthesia is a process well-documented in the anesthetic record. There are monitors attached to the patient that will alarm if there is any problem with such things as heartbeat, blood oxygen level, or a displaced endotracheal tube. There are several other persons in the operating room: one or more surgeons, a scrub nurse, a circulating nurse, often various others. All of these folks would also be aware of the alarms.
Your best bet would be to cause problems at the BEGINNING of the procedure: initial induction of anesthesia is a well-known high risk time – cardiac arrest and major drops in blood pressure being biggies. If you want your anesthesiologist to kill someone, that is the time to do it because it would be much harder to trace. You could sneak in additional (and lethal) drugs at the time of induction of anesthesia (such as potassium chloride or insulin) but simple blood tests would pick those up if someone thought to get them (and they would). And if a patient crashes during induction of anesthesia, the room immediately fills with many people and all sorts of things are immediately checked. Afterwards, the whole incident is gone over with a microscope. Operating room deaths are a very big deal and trigger major investigations at all hospitals by a committee, typically called M and M committees for “Morbidity and Mortality.” If a single anesthesiologist had more than one such death, the committee would REALLY check everything closely, and might even yank his/her privileges during the investigation.
All of this is good for patients but bad for novelists, I suppose. One way to make this work might be for the evil anesthesiologist to find a way to distract the other folks in the room or even get them to leave briefly. He/she could also destroy the blood samples that could implicate him. He/she would also have to falsify the anesthesia record somehow, which would also be difficult to do with people watching.