Related to, but not answered-by, this question (the answerer assumes, because many DMs rule that a monster is dead when it hits 0 HP, that Monsters don't get saving throws at all, which is not RAW, but a recognition/recommendation of a common convention).

The Nonlethal Rules text reads:

Sometimes an attacker wants to incapacitate a foe, rather than deal a killing blow. When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable.

However, the Instant Death feature reads:

Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.

Note that it doesn't specify what kind of damage, it just says, "if there's damage remaining."

This is usually not an issue in most parties, because, according to the PHB:

Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws.

(Emphasis Mine)

However, that is not a hard-and-fast rule, as some DMs roll saving throws for villains, especially if a character or player has qualms about killing, or for a recurring villain, important NPC, et cetera.

Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.

(Emphasis again Mine)

In these cases, the issue still exists. Let's say the Party is tasked with capturing a criminal alive to face justice. Throughout the fight, the Party manages to bring the fugitive down to 1 Hitpoint from his 20 HP maximum, and then the Party's Fighter goes to knock him out once and for all with his Great Sword... and crits. He rolls damage and ends-up dealing a staggering 22 points of damage.

Now, can the Fighter only knock-out the wanted criminal, or does the Party now have to figure-out how to convince the council that the convict is not, in-fact, dead, and simply "really, really knocked out," Arkham Asylum-style.