Bands like the Beatles, Modern Talking and many others were incredibly popular in the Eastern Block states, however you don't think of the Soviet Union in the same way as a major exporter of their culture to the west.
Did some bands reach popularity in the west in a similar way? If no, why not? And if yes, which come to mind?
I wouldn't say that Soviet rock music was ever popular in the mainstream of western consciousness, but I would say that there was definitely a bit of curiosity surrounding the music of the evil empire and that blossomed into some recognition in the west. I think this curiosity is best personified by the work of Joanna Fields who performed under the stage name Joanna Stingray; Fields is a Los Angeles-born American singer who reached middling success in the United States with her 1983 four-song album Beverly Hills Brat (the video of the single from that album featured a cameo by Andy Warhol).
A year later in 1984, Fields travelled to Leningrad (modern-day Saint-Petersburg) for a week and was inspired by her experience with the underground Soviet music scene, saying
[I was shown] an entirely different Russia-- here, people laughed, talked about everything, drank and played music. After those three days in Leningrad the only thing I could think about was how I could get back to the Soviet Union as soon as possible. [...]
The atmosphere at the rock club was the same as at concerts in Los Angeles. A couple of months later, [a friend] took me to a Mashina Vremeni concert in Moscow. The difference between an official concert and the rock club concert was like night and day. At the Mashina Vremeni concert, everyone sat stiffly in their seats and clapped between songs as if on cue. When a guitarist broke a guitar string in the middle of the performance, the concert stopped for 10 minutes while the guitarist replaced the string in total silence. Boris explained that the musicians were not allowed to do anything that did not conform to the approved program for the performance.
Soviet counterculture in the 1980s was mostly centered around nihilist irreverence with an undercurrent of righteous indignation. My personal favorite expression of this motif is the 1988 album Группа Крови (Blood Type) by the group Кино (Cinema). The album's name is a reference to the blood-group tattoo all Soviet soldiers received during service in Afghanistan to aid medics when blood transfusions were needed and the soldier was unable to express this information-- but I digress.
Joanna Stingray worked with the Australia-based Big Time Records' Los Angeles office to release the album Red Wave in 1986 which featured four bands from the Soviet Union (the first three of which are generally still revered as classics):
Stingray had to literally smuggle the recording used for this album out of the country. Red Wave was the first time Soviet rock music was released in the west-- unfortunately I couldn't find any concrete data about sales figures, but the album is still available through various vinyl retailers for around $25 and rated as 4.5/5 stars on Discogs. 10,000 were pressed and released for distribution. Stingray herself made this claim in the article I link at the end of this answer:
The album was received incredibly well in America. It had never occurred to anyone that Russia had rock music and rockers like in the rest of the world. It was a different view of Russia.
Joanna Stingray went on to marry the guitar player from Кино and release a handful of additional albums only in Russia. She also starred in some Russian films and music videos as well as assisting in the production of a fantastic documentary about Viktor Tsoi-- the lead singer of Кино whose life was cut short in 1990 by a car accident. His memorial in Moscow is still visited and adorned by fans in much the same way as Elliott Smith's Silver Lake analog.
This Moscow Times article features an interview with Joanna Stingray concerning her work and experience in the Soviet Union.
