The Delfini are eight points from safety in the relegation zone and facing an immediate return to the second tier - especially after a run of eight games without a win.
But former Giallorossi trainee Gianluca Caprari stunned his old club with the opener after 14 minutes and Roma's trademark attacking trident struggled to respond through the first half.
And though substitute Mattia Destro found a way through six minutes after the break, his colleagues Pablo Osvaldo and Francesco Totti endured a frustrating afternoon, with Federico Di Francesco almost snatching an away victory in the dying seconds.
Andreazzoli told reporters: "We can't allow ourselves to have such an attitude and I'm really disappointed this evening [Sunday].
"I did not think this could happen again, especially after what we have said in recent days. But it happened again.
"If you play this kind of game with a confident attitude then everything should fall into place.
"We have to give our opponents credit, they put us in difficulty and teams will do that when you're not playing at your best.
"We woke up in the second half and put on a good performance, for which I think we deserved to win."
Pescara assistant coach Bruno Nobili was understandably delighted with his side's resilient display in a partisan capital city.
He told La Repubblica: "It's such a shame, we could even have finished them off.
"We had the best chance of the whole game through Di Francesco at the end but I have to say (Maarten) Stekelenburg did well to prevent the goal.
"We had a pretty good game and held up will even though we were forced, inevitably, to defend. We did well not to lose our heads."