Pellegrini is no stranger to the later rounds of the Champions League having previously reached the semis with Villarreal and last season taken competition debutants Malaga to the Champions League quarter-finals but it is a first for City.
"I don't know if it's a relief but it's our first target for the year and I think to be in that stage is important for the club. I had a lot of trust that we could do it," Pellegrini told reporters.
"I'm very happy as this is an achievement that the club did not do in the last two years," added the Chilean.
"It is not easy to do it and two games before the group finishes we are now in the last 16 so that is very important for the club, the players and the fans, the whole club."
Under Mancini, City ended a 35-year trophy drought with the FA Cup in 2011 and added the Premier League title the following season but despite an expensively assembled team, they failed to make an impression in Europe and suffered two successive group-stage exits.
There has been an element of luck about City's progression after being drawn in a relatively easy Group D compared to last season's matchups against Real Madrid, Ajax Amsterdam and eventual finalists Borussia Dortmund.
ON FIRE
Some defensive question marks remain after the Manchester club were crushed 3-1 at home last month by holders and group leaders Bayern Munich and they conceded twice against an average CSKA side.
But the potency of City's attack is hard to dispute following Alvaro Negredo's hat-trick and Sergio Aguero's two goals which took him past Francis Lee as the club's top scorer in European competition with 12.
"Absolutely delighted that someone has FINALLY broken my Euro record, couldn't have lost it to a better player," former striker Lee, who played for City from 1967 to 1974, said on his Twitter account.
"Fabulous performance! Aguero is on fire, The Beast (Negredo) is a proper CF (centre forward) & (Samir) Nasri was brilliant. Historic evening!"
Pellegrini believes the performance against CSKA means City will be among the teams to be feared when the draw for the knockout phase takes place.
"We have a strong team so the minimum target was to qualify for the last 16, but it is not our only target, we must continue in Europe, we have a strong squad and we will see after the draw," he said.
"We don't know who we will have in February but I am sure that the team we play - they will not be very happy to play against us."
(Writing by Alison Wildey; Editing by Sonia Oxley)
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