The Croatia attacker's move to West Ham United was confirmed on Tuesday morning.
Vlasic has penned a five-year contract with the Irons after three impressive seasons with CSKA. The 23 year-old played more 250 senior matches, scored 33 goals and assisted 21 more for CSKA over the previous three years and was named Russian Premier League Footballer of the Year in 2020.
Despite his huge influence on CSKA's team, Tarkhanov could sense Vlasic was becoming increasingly unsettled this season as interest in his signing was firming up during the summer market.
Tarkhanov told Sport-Express: “Vlasic wanted to leave for a long time, but last year he at least played well, but this year he didn't look very good after the national team.
"Apparently, the thoughts were already about leaving. Naturally, there was no such zeal and charisma to play for CSKA. Does it make sense to keep the player against his will? There are no irreplaceable ones."
Indeed, Tarkhanov is confident CSKA can use the cash raised from selling Vlasic to West Ham to find a good replacement.
He added: "They also made money on him. With this money, you can buy a good attacking midfielder."
With Croatia, Vlasic has 26 senior caps and featured four times at the recent UEFA Euro 2020 finals.
- updated September 1