A better Soviet Moon Program, capable to almost go on their own to the moon, but has some flaws so it doesn't absolutely win the moon race, almost, and Soviet program, only scheduled to launch a week before Apollo 11, but halted due to some technical errors.
Apollo 11 goes as almost OTL, but the lunar lander are somehow failed, Armstrong and Aldrin stranded on the moon without much hope for return to Earth, Oxygen and water supplies only enough for three days...
Soviet Union goes to the rescue and willing to Help USA, at exchange of fixing that minor error, and due to emergency involved with two Americans stranded on the moon, NASA agreed to put away their differences, and half gambling to sent another rocket to the moon in a day and a half's notice.
The crew is just one soviet cosmonaut and one american astronauts, they minimize all non necessary supplies, and ordered to throw away scientific only instruments to save weight, so two US astronauts can be picked up, preferably alive.
With minimal fuss and celebrations, the "Duct Tape and Spit Rocket" are launched, and successfully rescue Armstrong and Aldrin alive (although hungry and dehydrated).
They arrived at earth, welcomed as Heroes, and from there, US-Soviet relations warmed to the point of friendliness...
From there, joint space missions become much more common.