City secured a place in the last 16 for the first time with a 5-2 win over CSKA Moscow at the Etihad Stadium.
Alvaro Negredo claimed a hat-trick and Sergio Agüero netted twice as City put the last two seasons' group stage disappointment behind them.
Pellegrini, who guided Villarreal to the semi-finals in 2006 and Málaga to the last eight last season, hailed a "big achievement".
"We didn't do it in the last two years - it's not easy to do it.
"Two games before the group is finished and we are in the round of 16. I think it is very important for the players, the fans, the whole club.
"There are some important teams in Europe but I think we have a strong squad and we'll see where we can go in this Champions League."
Agüero hit his first in the third minute from the penalty spot after David Silva was fouled and added a second on 20 minutes, before Negredo made it 3-0.
Seydou Doumbia pulled one back and also scored a penalty following Negredo's second, and the Spaniard headed in his hat-trick goal in the final minute.