I don't know if it's "the save" per se. the save became an official MLB stat in 1969. But during the 70's, many teams used their 'closer' for multiple innings. guys like Sutter, Gossage, Mike Marshall, Fingers, Eckersley, Lee Smith, etc would routinely work two or even three innings. Mike Marshall threw over 200 innings in relief in 1974. (admittedly, he was a unique character......)
the idea of the one-inning reliever is a more recent development.
it wasn't until the 1980's that the number of saves exceeded the number of complete games. by 1995, there were approximately 4 saves for every complete game.
of course, part of this is that teams used to have a 9 or 10-man pitching staff, so you could not afford to have 1-inning relievers. as teams moved to 12 and 13-man pitching staffs, it was more feasible to have a 'setup' guy, a closer, etc.
and - the rising importance of pitch counts has made a big difference. if the starter is limited to 100 pitches a game, unless that starter is really efficient, you are going to need more relievers.
(Mike Morris would like to say something - take your pitch count and shove it up your..........)