Two that come to my mind are “Max Headroom” and “Beauty and the Beast.” The production values of TV seemed to improve incredibly after these two series aired. Max only ran six episodes, but it had production values far beyond anything else on TV at the time. And Beauty followed soon after. Love them or hate them, you had to admit that they were beautiful to look at.
Actually, Monty Python only ran for 45 episodes. it was over a five year span, though. I think it’s safe to say that it completely changed both comedy and television.
Again, Police Squad in 1984 - no laugh track (but still studio bound) 
I would add Murder One, which only lasted 2 seasons, and IIRC, was the first show to have one major story arc last an entire season without any sort of resolution at the end of an episode (other than soaps). Now it seems that half of the tv shows (Heroes, Lost, Veronica Mars, etc. . .) are following suit.
I came into this thread to suggest Mozart, but if you lot would rather wallow in your vapid television programmes, I won’t be responsible!
The single episode of Turn-On that ran in 1969 is memorable just as what not to do, until such time as the audience came to tolerate goofy visual effects at microsecond pace, sometime in the eighties.
Sorry, I don’t necessarily pick up on the subtleties of sarcasm.
I thought of this, but recall the show was “James at 16” in its second year, making it not-all-that short-lived.
Damn, I feel old!
Nope.
Not that anyone paid attention, but **B5 ** went on the air 18 months prior to Murder One.
No doubt. Curb Your Enthusiasm and Larry Sanders preceded it, too. Yet it wasn’t until after AD that it became really commonplace. I think a lot of industry types noticed its passionate cult base and thought they’d see if they could emulate it.
Max Headroom had 14 episodes (although not all of them ran originally in the US). Why this show isn’t on DVD when Mama’s Family is is unfathomable.
Also, several shows in the 1980’s (the only one that springs quickly to mind is Wiseguy) had story arcs, and I suspect that there may have been examples of this form in earlier television (they had existed in radio).
Wouldn’t Twin Peaks qualify in that regard?
IMDB says 98 episodes, not 38.
*The Monkees * ran for two seasons and about 58 episodes (which is perhaps not quite as short-lived as the OP asked for). Most folks who were kids back then have fond memories of the show and the band, though most of us think of it as a guilty pleasure. Went into syndication, and kept winning new fans for the band decade after decade. The DVDs seem to be selling well, and the Monkees still have an active fan base today. Not sure how influential the show was–I can’t seem to think of any other show that was obviously influenced by it.
See post #8.
That’s what I was going to mention, and I do think it had a huge influence. On the one hand, it set the standard for bubblegum-rock kidvid from The Partridge Family to Josie & the Pussycats; on the other hand, the “romps” in each episode were seminal rock videos. Of course it was not totally original–it was obviously indebted to A Hard Day’s Night–but I think it had a big hand in shaping music television to this day.
And of course Mike Nesmith often gets credit for “inventing” the music video with Elephant Parts.
Exactly. It matters little that those other shows in the same style preceded AD. The industry loved AD (especially writers and producers) so that’s why shows in that vein keep churning out.
Much as I disliked it, I think “The Tom Green Show” was influential as an ancestor of “shock TV” series. “Jackass,” “Fear Factor” and “Punk’d” owe a lot to Tom Green.
max only ran six eps? 
wow. i thought it went way longer than that.
that show totally yanked me in. it’s where i finally started looking at computers as actually going to amount to something. coolest show ever.