This interesting Wikipedia entry says that Sweetwater was scheduled to be the first act at Woodstock but “problems within the band” — use your imagination — occurred and Richie Havens opened the historical festival. The LA-based jazz/rock fusion group followed. Thus, the item points out, Sweetwater was the first band to play. It’s a distinction.
This is Take it From the Splice, Boys. I’ve loved the first two minutes and the end of this song since I got the album in the early 1970s.