You're reading history in reverse. The Soviets landed a robotic probe on the moon years before the USA succeeded with human pilots,
https://en.wikipedia.org/wiki/Luna_9 ("Luna 9")
https://en.wikipedia.org/wiki/List_of_missions_to_the_Moon#M... ("List of missions to the Moon—Mission milestone by country")
You absolutely don't need sophisticated technology or giant budgets to accomplish this.