This is a difficult question to answer but I'll give it a shot. As mentioned earlier, many post-Communist countries have impressive education rankings in Math and Science today. For example, according to PISA, Estonia scored 520 in math, one of the highest rates in the world. Run a regression analysis comparing GDP per capita to PISA test scores, and many of the outliers will be post-Communist countries. For example, dirt poor Moldova scores 420, much higher than predicted by it's level of development. Take a look at the list of nobel prize winners by country, and you will see a large number number of people educated in the former Soviet Union in a diverse range of fields including Physics, Chemistry and Medicine. Of course, none of this really answers the question, but it does suggest that Soviet Union was doing something right when it came to education.

I think the first area where the USSR had real accomplishments was simply the massive expansion of access to education. In 1897 something like a fifth of the population of the Russian Empire was literate and literacy rates were especially low in Central Asia. One of the first acts of the government was to launch massive literacy campaigns. The Soviet Union managed close to near universal primary school education by the 1930s, and about one third of Soviet students attended high school by the 1950s. I wouldn't say the Soviet Union surpassed the West in expanding education access because western developed countries largely had universal primary schooling, but these advances are nonetheless impressive. The process of making education available was part of implementing Communist ideology, and the genereal Soviet state building process. Lenin's wife, Nadezha Krupskaya, a teacher, for example played an important role in shaping education policy. Schools were a place where the young were a captive audience, who could be indoctrinated with collective morals. But a lot of real education took place as well, and in some ways Soviet education could be highly progressive, with it's emphasis on gender equality and early opposition to harsh corporal punishment. While patriarchy never went away, I think it's important to point out about 40% of hard science PhDs were awarded to women in 1960s USSR.

The needs of the state played a powerful role in determining priorities of the USSR. Science (and therefore Science education) was seen as a geostrategic priority. First the race to nuclear weapons, and later the space race led to Soviet government to emphasize science training. Indeed, the launch of Sputnik forced a lot of soul searching in the US about the state of STEM education in America. Science, math and technical studies took up a much greater amount of the school day in the west (I see this referred to a lot, but I can't find a direct comparison of the average school day anywhere) and was the center of education. The USSR produced two to three times as many engineers and scientists per capita as the US, and at least in those areas of special interest to Soviet Union such as physics, Soviet scientists were right at the cutting edge.

However, there's some misleading ideas about the superiority of Soviet science floating around from the cold war era. The majority of Soviet engineers never worked as engineers or scientists but rather as government administrators and bureaucrats. One of the most obvious things about the USSR is that it was a communist state, and so there was no price mechanism or market incentives to govern the economy. Soviet planners instead did this with data and statistics. Even in the US, the government is one of the biggest generators of data and statistics, and this was double true in the USSR. The Soviet Union has often been described as the first technocracy because of the preponderance of engineers in high office, both political and administrative. Some pretty impressive achievements in statistical pattern recognition and data analysis were made by Soviet statisticians out of necessity to manage and understand all the data being created. More speculatively, I've read that Russia produces so many elite programmers today because the type of algorythmic thinking necessary for this kind of work translates easily to computer programming.

I've got to go do actual work now, but I hope my answer is useful.

Education and Professional Employment in the U.S.S.R. By Nicholas De Witt

Science in Russia and the Soviet Union: A Short History By Loren R. Graham

The Soviet Education Model : Russia’s Communist Legacy in Schools Past & Present

The Physics of Spin:Sputnik Politics and American Physicists in the 1950s, by David Kaiser