Neither side had previously managed to progress from the last 16 heading into the second leg and after a 1-1 draw in the first match in Moscow, Sevilla were regarded as favourites having secured an away goal.
Keisuke Honda's powerful free-kick saw CSKA through as Sevilla goalkeeper Andres Palop made a mess of what should have been a straightforward save in the 55th minute.
Luis Fabiano nearly ground out a goal in the 78th minute but failed to capitalise on the chance as Sevilla - needing to score twice - seemed to accept the inevitable.
"The result has been disastrous and we can only apologise to our fans. We had high hopes placed in the tie. The maximum and sole responsibility is mine and I will repeat it as often as necessary in this press conference," said Jimenez.
"I have lived with the criticism from you [the media] and others over time and have been respectful. I will keep working for the critics and for the non-critical. Manolo Jimenez has been given many kicks in the backside but I stand up to it all."