There's some subjectivity in where you consider it standardized, but I would put it later than that. Throughout the period when IBM's System/360 was popular, DEC's PDP series was also popular, which used 36-bit words and a somewhat varying definition of what constituted a byte.
If I had to pick a date, I'd pick 1977 as when it solidified. A few elements: ARPAnet standardized on octet-based protocols in the early 1970s, and was getting widespread by the mid 1970s; three popular microcomputers based on 8-bit architectures were introduced in 1977 (the Apple II, TRS-80, and Commodore PET); and DEC introduced the 32-bit VAX in 1977.
And thanks to good old SMTP originally not being specified as 8-bit, we get freakshows like UTF-7 http://en.wikipedia.org/wiki/UTF-7
I knew bytes were ambiguously sized, but I had thought a nibble/nybble was (A) always half an octet, and (B) a fun Apple coders magazine [0].
But sure enough, I was very wrong on (A): Apple II disk writes were done in "sets of 5-bit or, later, 6-bit nibbles" [1].
Thank you, chiph!
[0] http://www.nibblemagazine.com/ [1] https://en.wikipedia.org/wiki/Nibble#History
Before that you had CDC machines with 6-bit bytes (or Nybbles if you owned an Apple ][ with a floppy disk)