They really didn't. The Soviet space program operated with around a third of the US Space program (about one 200th of the USSR economy in 1970), or the equivalent of, give or take, 30 billion 2020 US dollars, or around 10 times the budget of SpaceX, notwithstanding technological price deflation.

With 1970s tech and only ten times the budget, the Soviet space program certainly innovated a lot more than SpaceX did, in all fields, from computer science to basic physics, to metallurgy, to jet engines, and built more ambitious missions.