The Giallorossi picked up all three points in the last minute of the game, thanks to Edin Dzeko's close-range finish.
Dzeko also scored his side's first-half equaliser, before Pellegrini himself gave Roma a lead that they could not quite see through to the end.
Dzeko restored the advantage at the last possible moment, however, sparking wild celebrations at the Benito Stirpe as the entire playing stuff rushed to mob the No. 9.
“It was vital that we won, we absolutely had to and I think you can see that in the way we celebrated," Pellegrini said.
"It’s a tough place to come, they very aggressive on a small pitch. We showed that we will keep on fighting, and in the end we got the result.”
He added: “Dropping two points would have been a big disappointment for everyone. We go into the derby with a few good results under our belt.
"We know it will be a very tough game again, they will be fired up for it – but so will we.”
Pellegrini celebrated his own goal by sucking his thumb - a gesture he confirmed afterwards had a special significance.
“The desire we had to win was important in getting there in the end, and it was amazing to be celebrate with the entire bench," he said.
"We are really a close-knit group, battling right until the last moment to achieve what we can.
"My goal celebration was for my wife, who is expecting. I want to dedicate this win to my little girl, who will be arriving quite soon.”
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