Strictly speaking the post is not about 10 best cosmonauts but about cosmonauts involved in the first 10 successful Soviet space flights. Starting from the seventh the flights were not more individual so you'll find here more than 10 persons. Let's start.
1. Yuri Gagarin
First flight of a man into the space at the height of 327 kilometers lasting in total 108 minutes that have changed his life and life of our planet.
2. Gherman Titov was Gagarin's substitute for any unforeseen circumstances and afterwards was chosen to be the second Soviet man in space in 1961. He still holds a record - he was a youngest man ever been in space. At the moment of the flight he was 25.
3. Andriyan Nikolayev participated in the first joint flight of two space ships Vostok-3 and Vostok-4. For that time his flight was the longest - 94 hours and 25 minutes.
4. Pavel Popovich was the second cosmonaut of the mention first joint flight. He was the pilot of Vostok-4 and approached the partner space ship for 6 km.
5. Valeriy Bykovsky piloting Vostok-5 was one of the second joint flight's pilots. He set up a new flight endurance record - 119 hours and 6 minutes. It remains for this day because there have never been longer solo flights.
6. Valentina Tereshkova was not just the first woman in space but she was the one who approached the partner space ship (Vostok 5) at 5 kilometer distance.