It actually started during the second World War, with Klaus Fuchs in the Manhattan Projekt. Other examples are the russian space shuttle buran that doesn't seem to differ much from the american one and the russian akula class submarines, which are sometimes even called walker class, named after John Anthony walker, another soviet spy whose spying influenced their development to a large degree.
But are there any cases where the soviets had technology that was vastly superior to the Western one? And on top of that, did the West ever successfully steal some?
My Cold War tech I can comfortably talk about is limited to tanks as a heads up.
I can’t necessarily say it was stealing but in terms of armor and tanks, the US frequently copied many Russian designs to some extent. The most notable is the large, cast, hemispherical turrets signature to the M48s and M60, which were a result of direct inspiration from the T-54/55 and IS-3 as those tanks, specially the IS-3, terrified the western allies. The Russian signature pike nose design of many of their heavies was also recreated to some extent using the cast, rounded hull on the M48 and M103.
Other specific designs include the narrow face mantlet, much like how the Germans began to copy late in WW2. The mantlet as seen on the T-54/55, T-62, and T-64’s, caused the Allies progressively moved towards smaller, more internalized gun mantlets as seen in something like the British Chieftain or even later tanks such as the M1, Challenger, and Leopard 2.
A few minor composite armor designs were copied by the West as well. Not necessarily composites like chobam which has been being experimented on by both the East and West since the 50’s, but more so composite casting.
For example, early models of the T-64 had a dual casting system for the turret in which a cast steel “tub” for the turret face was formed, and then had molten aluminum poured into it to increase defense against shapes charge warheads. The Allies later copied this on some appliqué armor packages using various soft steel/high hardness steel like on later Leopard 1s.
There are a number of documented examples of technology flowing from East to West.
Perhaps most famously, Pyotr Ya. Ufimtsev's rather obscure academic paper "Method of Edge Waves in the Physical Theory of Diffraction" was rediscovered by a then 36-year-old Lockheed mathematician and radar specialist, Denys Overholser, and served as the critical "a-ha!" moment for the development of truly low observable aircraft, most famously the F-117 and B-2:
“The nuggets Denys unearthed were found near the end of its forty pages. As he explained it, Ufimtsev had revisited a century-old set of formulas derived by Scottish physicist James Clerk Maxwell and later refined by the German electromagnetics expert Arnold Johannes Sommerfeld. These calculations predicted the manner in which a given geometric configuration would reflect electromagnetic radiation. Ufimtsev had taken this early work a step further. “Ben, this guy has shown us how to accurately calculate radar cross sections across the surface of the wing and at the edge of the wing and put together these two calculations for an accurate total.” (Ben R Rich and Leo Janos. “Skunk Works: A Personal Memoir of My Years at Lockheed.”)
In the December 16, 1986 NY Times article "Technology from Russia finds a Niche in U.S. Life", Malcolm Brown documents a number of licensed technology transfers from the Soviet Union to the United States. These include welding process improvements, electromagnetic metal casting, and various coating technologies.
Some Redditors reading this post may have benefited from the 1974 invention of Radial Keratotomy by a Russian opthamologist, Svyatoslav Fyodorov (Lam, "Saving Sight: An eye surgeon's look at life behind the mask and the heroes who changed the way we see"). Fyodorov's work was so pioneering that RK was known in the West as the "Russian Technique" for some time after its introduction.
Perhaps of dubious benefit, Western intelligence agencies recognized the clear Soviet lead in psychic and paranormal research. I will leave it to you, Agent Mulder, to come to your own conclusions on whether these were rubles well-spent (as well as the CIA and DIA monies spent on the West's similar efforts):
https://www.cia.gov/library/readingroom/docs/CIA-RDP96-00792R000500210001-9.pdf
https://www.cia.gov/library/readingroom/docs/CIA-RDP96-00787R000500420001-2.pdf
https://www.cia.gov/library/readingroom/docs/CIA-RDP96-00792R000500730003-0.pdf
A type of liquid-fueled rocket engine was pioneered in the Soviet Union and later exported to the West. So-called closed-loop kerosene engines took advantage of Soviet advances in metallurgy and design methodology to outperform many Western designs, and are still used today by Western launch vehicles (note that there are some political considerations relating to ballistic missile non-proliferation attached to some of these procurement deals, but that should not take away from Soviet achievement in this arena). References include http://spacenews.com/36232space-propulsion-development-of-us-closed-loop-kerolox-engine-stuck-in/ and the documentary https://www.youtube.com/watch?v=TMbl_ofF3AM.
EDIT: added Tokamaks —
The Tokamak, a type of magnetic confinement scheme invented in the Soviet Union, is a leading approach for fusion reactor design and now widely used by Western research institutions. (https://www.iter.org/mach/tokamak)