With slide-rules, no less, to some extent.
NASA had a bit of computing power back then. When Apollo 13 had a fuel cell explosion on the way to the moon, they quickly realized that any return flight would have to include the lunar lander for life support. The return flight required a critical rocket burn on the far side of the moon. Too long a burn would miss the earth, and too short a burn would overshoot it. Unfortunately, NASA had never calculated this burn with the mass of the lander remaining attached to the command module.
Getting the right numbers in time required massive computing power. NASA literally confiscated computing time from all the available IBM 360 computers in the country. It involved jetting boxes of punch cards around the country and running them on volunteered time from the universities and businesses that offered them. Your smart watch has far more computing than NASA had at the time. The burn worked, and all returned home safely