A Soviet anti-satellite system based on a killer satellite

Did the Soviet Union have an equivalent to the American anti-satellite missile programme?

Yes; the Soviet approach is described as a converted ICBM that placed a killer satellite against its target, possibly before the American system.

The answer places Soviet anti-satellite development at least alongside, and possibly before, the American effort. Its method is described as a converted intercontinental ballistic missile carrying a killer satellite. This gives the Soviet Union a counterspace option of its own during the Cold War. The dating is presented from memory, so the useful takeaway is the existence and broad form of the system rather than a precise claim of priority.