Pellegrini converted from the penalty spot after just six minutes at the Stadio Olimpico, capitalising in nerveless fashion after Paulo Dybala had been brought down inside the box.
That finish ultimately proved decisive; Chris Smalling and Tammy Abraham producing two brilliant defensive efforts in injury time to ensure the clean sheet that was enough to give the side victory.
Coach Jose Mourinho - suspended from the touchline for this one - sprung a surprise or two with his starting line-up, as Benjamin Tahirovic started his first game for the club and Dybala partnered Nicolo Zaniolo in attack.
It was Dybala who sparked the game into life, as he latched onto a clever ball into the box before being clumsily brought down by the covering defender.
After Pellegrini had put the chance away the Giallorossi continued to create chances, with Dybala himself going close and Zaniolo denied at close range by Bologna goalkeeper Lukasz Skorupski.
In the second half Bologna pushed on more, needing an equaliser, but they were fended off with a measure of comfort by the hosts - at least until the closing minutes, where substitute Abraham was on hand to head a goalbound effort away off the line.
The Giallorossi are back in action on Sunday as they travel to face AC Milan.
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