This phrase is properly punctuated with quotation marks: Live from New York, it's "Saturday Night!" This is because it dates back to the very first show, a time when the official title was "NBC's Saturday Night." The phrase has never changed out of tradition, out of inertia, and out of the realization that using the word "live" twice in the same sentence would be redundant.
During the first season, Chevy Chase got to say the magic phrase in all but two of the episodes; since then, the duty and privilege have been spread more evenly among the cast members (plus the occasional guest host).
In addition to the examples above of when the phrase was not used, one particularly memorable example occurred in the February 23, 1991 show, which opened with a "McLaughlin Group" parody that ended with Dana Carvey as John McLaughlin asking each panelist in turn, "How do we start the show?" All responded in turn, "Live from New York, it's 'Saturday Night.'" Carvey, of course, proclaimed them all WRONG!, faced the camera, and said, "Show, show, show, here we go!" Cue music, cue opening titles, cue Don Pardo.