In a first half littered with fouls and stoppages, Roma carved out the best opportunity, as Bryan Zaragoza dragged wide from 18 yards on the 20-minute mark following a superb counterattack led by Donyell Malen.
Bologna offered little in response, but Mile Svilar was called into action just two minutes before the break, as the Serbian stopper sharply dashed across his line to repel Bernardeschi's 25-yard strike.
However, the pattern of a slow-burning first period was thankfully not repeated after the restart, as Bologna burst into life on 49 minutes.
Jonathan Rowe's clever pass on the edge of Roma's penalty box freed up Bernardeschi in the right channel and the Italy international curled an unstoppable angled drive past Svilar.
Malen looked to lead the Roma revival from there as the Dutch forward arrowed a close-range strike narrowly past Łukasz Skorupski's near post just four minutes later before Svilar flew off his line to bravely thwart Tommaso Pobega on the hour.
As the contest opened up from there, Roma seized the chance to equalise via the quick-thinking Pellegrini.
Captain Bryan Cristante forced an error from Joao Mario down the visitors' left flank, and his through ball allowed Malen to test Skorupski, but with the Bologna back line slow to react to the loose ball, Pellegrini calmly slotted home from 10 yards out.
That shook up the game in the closing stages, and Malen came agonisingly close to pinching a smash-and-grab win on 76 minutes as his miscued toe poke was deflected on to the post by Jhon Lucumí before Bologna sub Charalampos Lykogiannis bulleted a header against Svilar’s crossbar on 90 minutes.
A draw for Bologna extends their unbeaten UEL run to 10 games, after losing out to Aston Villa in their league phase opener, and Vincenzo Italiano's team are also unbeaten in five home clashes with Roma across all competitions, ahead of a crunch second leg in Rome on March 19th.
Flashscore Man of the Match: Federico Bernardeschi (Bologna)