A 2-1 win in Rome was undone as the Portuguese outfit matched that result in 90 minutes to take the game to extra-time before Alex Telles scored a penalty that sent them through and avoided the lottery of a shootout.
Italian Serie A club AS Roma said they had parted company with coach Eusebio Di Francesco on Thursday, one day after being knocked out of the Champions League by Porto.
A semifinalist one season ago, Eusebio Di Francesco will reportedly be fired by i Lupi after coughing up a 2-1 first leg lead in Portugal on Thursday.
In extra-time, Dzeko wasted a succession of chances to score the goal that would likely have taken Roma through.
Porto's lead lasted barely 10 minutes as Eder Militao lunged in on Argentine Perotti, with De Rossi coolly rolling his penalty past three-time Champions League victor Iker Casillas.
Pallotta posted to social media: "Last year we asked for VAR in the Champions League because we got screwed in the semi-final and tonight, they've got VAR and we still get robbed".
However, Roma were handed a lifeline after being awarded a penalty, which captain Daniele De Rossi duly converted to put the Serie A side back ahead in the tie.
Blues owner Roman Abramovich is set on Champions League qualification next season and that continues to remain in the balance.
Former Roma midfielder Di Francsco helped the team to their last Serie A title as a player in 2001.
Ranieri coached Roma for three seasons from 2009 to 2011.
The 67-year-old Italian's brief spell as manager of Premier League strugglers Fulham ended last month after only four months in charge.
R. Kelly jailed until he pays $161K in child support
Drea was among multiple women who alleged sexual misconduct against the singer in the recent Lifetime docuseries " Surviving R. After Kelly's interview aired, her parents held a press conference saying they were "disturbed" by his comments.