The Russian Premier League side were beaten 2-1 in the Czech Republic on Tuesday, having been ahead after 65 minutes through Ahmed Musa.
Ultimately, they were denied a spot in the Europa League thanks to goals in the last 14 minutes from Daniel Kolar and Tomas Wagner, with Dzagoev's dismissal after tangling with Vaclav Prochazka off the ball two minutes after the CSKA goal proving crucial, according to Slutskiy.
"The match took on a different character when we went down to 10 men," he said.
"We didn't have the strength and concentration to overcome the resistance of our opponent. A key factor was the removal of Dzagoev."
Slutsky also revealed that Keisuke Honda, who supplied the cross for Musa to score the opener, was held back from starting the game due to a knee injury.
The Japan international was brought on seven minutes after the break, with Slutskiy adding: "He had a sore knee, and we decided that he was unlikely to survive a full game."