Sir Bernard Lovell, the astronomer, was among the team listening to transmissions coming from the area of space and began tracking the unmanned Soviet spacecraft Luna 15, which was trying to collect samples of lunar soil and rock and then return to Earth before the US mission.
The recordings from Jodrell's Lovell radio telescope, which were hidden in archives until researchers found them, show the Russian craft orbited the Moon and crash-landed onto its surface at 15:50 on July 21 – just a few hours before the Americans lifted off.
In the newly released recordings, which were made over three days, Sir Bernard, the founder of Jodrell Bank, can be heard narrating events with conversation from the Apollo 11 astronauts in the background.
Sir Bernard notes a change in the orbit of Luna 15 to take it closer to the US landing site and later reports a rumour from a "well-informed source in Moscow" that the craft is about to land.
People in Jodrell's control room can then be heard shouting "it's landing" and "it's going down much too fast" as they track Luna 15's final moments before it crashes.
A voice is later heard saying: "I say, this has really been drama of the highest order."
The recordings have been released by The University of Manchester's Jodrell Bank Centre for Astrophysics to celebrate the 40th anniversary of the Moon landings.