First of all, in Russia, the term 'spacesuit' (скафандр) can be a little ambiguous as it refers not only to space suits, but to generally any suit that isolates a human from the environment and provides a living environment within. Going by that characteristic, the first closed- loop suits were developed very early on in the 1930s. These models (СК–ЦАГИ and Ч) had a closed-loop ventilation system already since the goal was to allow a prolonged stay in the stratosphere. The need for developing hermetically sealed suits led to the creation of the "Zvezda" R&D facilities.
"Zvezda" focused on development of different suits and egress mechanisms. Importantly for this question, they were tasked with the development of the pod for the animal launches into space (which Laika flew in), as seen below.
Technically, this is the first spacesuit with a closed-loop ventilation system, but this is stretching the definition of the word "spacesuit" a bit. So let's look at the spacesuits made for humans.
They were also made by "Zvezda". Indeed, up until "Berkut", the spacesuits were all open-loop. However, development of the closed-loop spacesuits commenced in 1966, a year after Leonov's EVA. Two spacesuits were developed: "Yastreb" (hawk) and "Krechet" (gyrfalcon), also known as "Krechet-94".
Yastreb (shown below) was intended for EVAs in orbit. It was used only once, when Khrunov and Alekseev moved from Souyz-5 to Souyz-4 though outer space in 1969.
"Krechet" (shown below) was intended to be used by the Soviet moon program. It had a much more complicated design, including food/drink systems, waste processing, radio, telemetry. It had passed all the tests in 1969, but never saw any use as the moon landing program was cancelled.
The direct descendant of the "Krechet" is "Orlan" (sea eagle), which was designed for use on the space stations and is used in the ISS to this day.
To summarize: the first Soviet spacesuit that had closed-loop ventilation and saw any use was "Yastreb". "Krechet" was developed alongside it and had completed all the tests in the same year as "Yastreb" was first used in an EVA, but it never was used in space.