Kroos' dreadful backpass allowed Nikola Vlasic, on loan from Premier League Everton, to put CSKA ahead.
Arsenal quickly ended any doubt over the contest on the field as Aaron Ramsey and Alexandre Lacazette both scored twice in the first half to surely put the tie out of CKSA's reach.
Wayne Rooney's 79th minute strike ended Manchester United's goalless streak and Louis van Gaal was a happy man to avoid a fourth consecutive 0-0 draw.
Manchester United were on course to record a fourth consecutive goalless draw for the first time in their history as their scoreless run ticked into its 404th minute when under-fire skipper Wayne Rooney met Jesse Lingard's cushioned pass with a thumping header in front of the Stretford End.
It was former Monaco striker Anthony Martial's first Champions League goal for his new club and fifth this season since a high-profile transfer from the Ligue 1 club.
Manchester City had ended a three-game winless run by beating Manchester United 1-0 on Sunday, but they produced a strangely insipid display on Wednesday in a 2-1 home defeat by Russian champions CSKA Moscow in the Champions League.
UEFA have ordered CSKA Moscow to play the game behind closed doors because of racist behaviour by their fans - something Manchester City have direct experience of after Yaya Toure was on the receiving end of verbal abuse when the clubs met last season.
Premier League champions Manchester City F.C. go to Russia to face CSKA still looking for their first win in the UEFA Champions League this season. Chelsea, Bayern Munich, Barcelona and Paris Saint-Germain will also be in action on Tuesday.
Bayern Munich defeated Manchester City in their opening Champions League match and would now hope to maintain the momentum against hosts CSK Moscow.
Roma last featured in the competition in the 2010/2011 campaign when they reached the round of 16. The team will now host injury-hit CSKA Moscow.