The Russian team were surprise winners at the Sanchez Pizjuan on Tuesday but well worth their 2-1 victory on the night, which put them in the quarter-finals 3-2 on aggregate.
Sevilla left the pitch to whistles and chants for Jimenez to go from some sections of the home crowd, who have been unhappy for some time at what they believe are his conservative tactics.
"Of course it hurts. Now, it will be said it was the whole stadium, though it clearly wasn't," Jimenez told an increasingly tetchy news conference after the match.
"I'm the only person responsible for what has happened. CSKA are a very good team even though they don't have the name of other sides, and have waited for their chances and made the most of them."
Sevilla had high hopes of challenging for the La Liga title and of making the last eight of the Champions League this season with a squad that boasts one of the strongest attacking line ups in Spain.
A club record swoop for Alvaro Negredo, for a reported fee of 15 million euros from Real Madrid last year, added the young Spain striker to the proven talents of Luis Fabiano and Frederic Kanoute up front.
In midfield, they have the pace of players such as Jesus Navas, Diego Perotti, and Diego Capel. But on Tuesday the side were cautious in their approach and Jimenez started with only one natural striker, Luis Fabiano.
Sevilla lie fourth in La Liga, 21 points adrift of leaders Real Madrid, and are in the King's Cup final, but club president Jose Maria del Nido fell short of giving Jimenez a ringing endorsement.
"All I can say is that Jimenez will be coach until the end of the season," he told Spanish state television.
"We can't win the league so we have to concentrate on claiming third or fourth place and on winning the Cup."
