I think you have to separate natural sciences from maths here, and go farther back. Mathematics was Imperial Russia's strongest academic area well before the Soviet era.

Otherwise, in the realm of science, Russia was pretty much a backwater until the Soviet period, despite the efforts of rulers like Peter and Catherine the Great. In terms of 'household name' scientists, Russia had Lomonosov in the 18th century, and Mendeleev and Pavlov in the 19th. (and the latter is almost 20th) The situation in mathematics is very different though - Russia had a number of great mathematicians in the 19th century - Lobachevsky, Chebyshev, Lyapunov, Markov and Steklov for instance.

To see why the situation was so bad, consider the situation in Russia around 1700: There were very few books, both due to skepticism of the practice, shortage of paper, limited literacy and also limited interest. The entire country only printed about a half-dozen titles a year. -About as much as a single printing shop in a western European city might do. Russia was quite isolated from western learning, not least since works were written in Latin and the whole tradition ultimately had its roots in Catholic monasticism - another reason for conservative Russians to be a bit averse. So Russia was really in a very poor position to get in on the Scientific Revolution that was starting to the west. But the country did open up (a bit at least) to the west, starting with Peter the Great and attitudes started to change. An Academy of Sciences like western countries were doing then, and so on.

In 1703 Russia, under Peter and with his personal support, Russia finally prints its first secular book. (a good 250 years after Gutenberg) - and it's a mathematics book. Арифметика (Arithmetic) by Magnitsky. It becomes the main Russian math textbook for well over a century after that. Nevertheless Russia did not produce any great mathematicians yet in the 18th century, although Magnitsky's book was credited by Lomonosov as catalyzing his development (even though I can tell you Lomonosov's body of work contains very little actual mathematics though - even when he claims to be using it!)

Probably the most significant event in the development of mathematics there was at the end of the century, during Catherine, when S:t Petersburg managed to attract a group of mathematicians from Basel in Switzerland - Nicolaus and Daniel Bernoulli and Leonhard Euler - who were some of the best in the world at that time.. indeed some of the greatest of all time! So even though the talent was imported, suddenly Petersburg was a world center in mathematics. Another foreigner they attracted was Johann Bartels - who was not a hugely significant mathematician in his own right, but since he was a teacher of both Gauss (another short-lister for greatest mathematician ever) and Lobachevsky, he must've been doing something right! And if nothing else he provided connections to the important people and greaterr world of mathematics. In present day buzzwordology they had a bit of a "Knowledge Cluster" going on. The aformentioned Chebyshev was a teacher of Markov and Lyapunov. Lyapunov taught Steklov, and so on - a tradition was established.

They were not as successful at attracting talent within the natural sciences. Part of this is that around 1800, Russia was still far behind on the Industrial Revolution. In fact it's usually considered to have only reached Russia by the late 19th century. But much of the physical science of that era evolved out of it. Most of Europe's prominent chemists of the era were deeply involved in practical matters such as mining/geology/smelting/metallurgy or producing dyes and so forth, while in physics, the field of thermodynamics was developing out of the desire to improve the steam engine. Which of course requires you to have steam engines to study. Russian science responded by eschewing practicality and instead highlighted the 'purity' of theoretical work (see the first chapter of Kojevnikov for more on that).

So Russia was already good at mathematics since the late 18th century, had extended that to an extent to chemistry in the late 19th century, and managed to become a player in theoretical physics under Stalin. (and Lev Landau. His importance to Soviet physics cannot be overstated)

Kolmogorov reformed mathematics education in the Soviet school system in the 1960-80s era. One might compare to the 'New Math' in America. The content was different and less radical in the Soviet Union but they originated from the same impulse of perceived necessity in the space age. They did have some novelties in common though, like introducing set theory - which ended up being one of the most disliked aspects in both cases too.

Karp & Vogeli Chapter 3 goes into detail on Kolmogorov's reforms, but the short of it is that is is simply an education reform and overhaul of the curriculum (such as allowing elective math classes) not really a "system" or special pedagogy. Karp & Vogeli assess the reforms as a 'relative failure' though (also like New Math). Part of their summary states:

For students interested in mathematics, the reform was a positive phenomenon; however, its aim — to construct a conceptually rich and at the same time universally accessible school course in mathematics — was not achieved. The knowledge and skill levels of a considerable part of the students were lower than expected.

But they also make it clear that it did in fact lead to some lasting improvements as well.

(Main) References:

Karp, Vogeli: Russian Mathematics Education (2010)

Kojevnikov, Stalin's Great Science (2004)