Most references state that Sarah Michelle Gellar‘s first appearance as an actor was in the made-for-TV movie AN INVASION OF PRIVACY, which first aired on CBS 12 January 1983. The making of this film may well have been her first acting job, but her first appearance on television was as a guest star on the NBC sitcom LOVE, SIDNEY.
LOVE, SIDNEY was the first American TV show to feature a gay (if closeted) man as the main character, and starred the late Tony Randall, Swoosie Kurtz (now Joyce Flynn in MIKE & MOLLY), and a young Kaleena Kiff (last seen playing a female victim in SUPERNATURAL in 2008). Gellar appeared in Episode #25 titled “Sidney’s Spree” which was first shown on 10 October or 16 October 1982 (depending on whom one consults).
There is almost no video of LOVE, SIDNEY available on the web, and the series is not on DVD or VHS so far as I can determine. The rights to the show seem to be owned by Warner Bros.