1955 Brooklyn Dodgers.
Nope.
Well, it may not be the answer you were expecting, but the 1955 Dodgers did it, starting Don Newcombe, Billy Loes, Johnny Podres, Carl Erskine, Roger Craig, and Karl Spooner. If there is more than one answer, your question is pretty pointless.
I stand corrected. I was watching the Series I am referring to, and made the mistake of believing the announcers when they said it was the first time. I know it hasn't happened since.You're right. Sorta makes the question pointless. But I'll wait to see if anyone knows the 2nd time.
'71 Pirates
This is what I was originally looking for, but as HH has pointed out, I committed the cardinal sin of trivia and asked a question of which I did not know the answer.