Sputnik did broadcast the first signal from space. But the first voice... that was something else. Sputnik 1 was launched October 4th 1957. As Americans that's just the sort of thing that gets our gall. We had to keep up with the Jones'. I wrote about that first sputnik broadcast here. Let's cover what happened next.
Well November 1957 Russians launch Sputnik 2 with a small terrier on board. This went over about as well as Sputnik 1. We didn't even get around to forming NASA until July 28th 1958. So until then we had to rely on the "Ballistic Missile Agency." Not impressive. May 15th 1958 they launch Sputnik 3, an actual scientific instrument. This sent President Dwight Eisenhower into conniptions.
So to wrestle any degree of success we needed a "first." When we launched Explorer 1 on Jan. 31, 1958 that helped. It was a much more ambitious scientific project. But we weren't done. The fear was that these satellites were leading up to spy satellites, which of course they were.
NASA launched a communication vessel known as SCORE. The purpose of SCORE was to tape messages that were transmitted from earth at 150 MHz, and then send them back to earth. Dec. 18th, 1958 we launched SCORE into orbit via an Atlas rocket. It was a communications satellite, the first of it's kind.
Project SCORE first broadcast on December 19th, 1958 and carried a tape-recorded a Christmas greeting from President Dwight D. Eisenhower. It was broadcast on 107.97 MHz ad 107.94 MHz. Some texts will call thsi the first broadcast from space, but with the three preceding Sputnik's that is clearly not the case. But it is almost certainly the first voice broadcast from space.