Early editions of D&D direct that a character dies when reaching 0 hp, that is, instantly.
Hit Points and Hit Dice Any creature reduced to 0 hit points (or less) is dead. Page B6 (D&D Basic Box Set)
When was it first published as an option that characters do not die instantly in the general case?
In AD&D 1e, the PHB stated
Damage is meted out in hit points. If any creature reaches 0 or negative hit points, it is dead.
However, the DMG stated otherwise.
When any creature is brought to 0 hit points (optionally as low as –3 hit points if from the same blow which brought the total to 0), it is unconscious. In each of the next succeeding rounds 1 additional (negative) point will be lost until –10 is reached and the creature dies. Such loss and death are caused from bleeding, shock, convulsions, non-respiration, and similar causes. It ceases immediately on any round a friendly creature administers aid to the unconscious one. Aid consists of binding wounds, starting respiration, administering a draught (spirits, healing potion, etc.), or otherwise doing whatever is necessary to restore life.
Any character brought to 0 (or fewer) hit points and then revived will remain in a coma for 1-6 turns. Thereafter, he or she must rest for a full week, minimum. He or she will be incapable of any activity other than that necessary to move slowly to a place of rest and eat and sleep when there. The character cannot attack, defend, cast spells, use magic devices, carry burdens, run, study, research, or do anything else. This is true even if cure spells and/or healing potions are given to him or her, although if a heal spell is bestowed the prohibition no longer applies.
If any creature reaches a state of –6 or greater negative points before being revived, this could indicate scarring or the loss of some member, if you so choose. For example, a character struck by a fireball and then treated when at –9 might have horrible scar tissue on exposed areas of flesh — hands, arms, neck, face.
In AD&D 2e, the game went back to the 0 HP = death, according to Korvin's answer. And in 3.x, we went back to the idea of bleeding out and negative HP, although the specifics on how to heal, as well as the consequences, differed.
Fourth edition has introduced the concept of Death Saving throws as you know it in 5e. After reaching 0 HP, the character makes a death saving throw at each of its own turns, and after failing 3 times, they die.
However, the BECMI version1 had an Optional/Variant Rule that was similar (but harsher) to the early Death Saving throws from 4e in p. 266: (thanks to Korvin again for the quote)
For instance, when a character is reduced to 0 hit points or below in combat (or from death spells), he's not yet dead. He's unconscious and mortally wounded; if left untended, he will die. He must make a saving throw vs. death ray every turn. He makes the first roll on the round he drops to 0 hit point; he makes another every round he takes additional damage, and every 10 minutes (one turn) in addition. If he ever fails a roll, he's dead. If he keeps making his rolls until reached by a healing cleric, someone with the Healing general skill, or someone with a healing potion to get to him, he can be saved. If the healers can heal him up to 1 hit point or more, or the Healing skill roll is made at a penalty of —5 (regardless of whether it heals him up to positive hit point or not), then the character is alive. He's critically wounded—but he'll survive.
I assume this is where the "death ray" confusion present in the question earlier came. Note that these death save throws did not allow the character to stabilize themselves, they required external healing.
1 BECMI stands for Basic/Expert/Companion/Masters/Immortals, one of the D&D Basic Sets, by Mentzer.
