Andreolli scored with almost the final kick of the game as he powered home a shot from inside the box off the underside of the crossbar to break Fulham's hearts.
Fulham looked to be heading for a famous win when Brede Hangeland put them in front after 24 minutes as he rose highest to head home Bjorn Helge Riise's corner.
But the Cottagers were reduced to ten men 14 minutes from time when Stephen Kelly brought down John Arne Riise in the box to concede a penalty.
Confusion reigned at the sending off, with the referee appearing to show the red card to Hangeland before finally changing his mind and ordering off Kelly.
Mark Schwarzer came to Fulham's rescue as he saved the resulting penalty from Jeremy Menez with a superb stop, low down to his right.
However, Roma were not to be denied and Andreolli struck at the death to spark wild celebrations amongst the visitors' bench.
Roma, 11th in Serie A, arrived at Craven Cottage without talisman Francesco Totti and found themselves in desperate need of his inspiration during a difficult opening.
Marco Andreolli scrambled a dangerous cross by Bjorn Helge Riise to safety before keeper Doni had to make a sharp save off Jonathan Greening's shot.
Greening saw a second attempt deflected for a corner and Doni remained busy as he moved to gather a curling strike from Kelly that was destined for the bottom left corner.
Paul Konchesky threaded a magnificent ball through to Bobby Zamora but with Andreolli tugging at his shirt, he was unable to create the space to pull the trigger.
Roma were finding Fulham's crisp passing and movement a handful and when they gained possession themselves they were soon powered off the ball.
DominanceGiven their dominance, it came as little surprise when Hodgson's side surged ahead in the 24th minute.
Bjorn Helge Riise sent a hanging corner to the rear of the box where Hangeland escaped the grappling arms of Andreolli before outjumping the Roma defender to head the ball past Doni.
Roma issued a reminder of the threat they can pose when John Arne Riise - playing against his brother Bjorn Helge - unleashed a blistering 18-yard drive that drew an excellent save from Schwarzer.
Zamora was mobbed by Roma defenders as Fulham pressed ahead once more but the tempo had dipped after the goal.
The Italian club were spending more time on the ball but lacked a cutting edge in the final third.
Whatever Claudio Ranieri said during his interval team talk had the desired effect as Roma started the second half with far greater purpose.
OffsideThey even had the ball in the net in the 53rd minute but Daniele De Rossi was clearly offside when he nodded Philippe Mexes' shot into the net.
Substitute Simone Perrotta dragged a good chance wide as openings appeared in Fulham's defence, but the visitors were then hit on the counter.
Zamora raced down the left wing and directed a long ball across the goal intended for Diomansy Kamara only for Mexes to intervene at the crucial moment.
Doni did brilliantly to pluck the ball from Kamara's feet after Zamora had engineered a one on one - had he been a fraction later Fulham would have gone 2-0 ahead.
Schwarzer then saved Menez's penalty as Fulham fans were subjected to a
nerve-shredding finale that ended in agony.
Andreolli stole in unmarked at the far post and lashed home from six yards out with the last kick of the game.
FulhamTeam StatisticsRoma1Goals111st Half Goals04Shots on Target53Shots off Target84Blocked Shots1011Corners819Fouls119Offsides40Yellow Cards11Red Cards074.3Passing Success77.226Tackles1865.4Tackles Success66.741.9Possession58.145.7Territorial Advantage54.3