Mark Schwarzer had saved a late penalty from Jeremy Menez which also saw Stephen Kelly sent off for an innocuous tackle on John Arne Riise.
Fulham took a first-half lead through Brede Hangeland, who was initially red carded for Kelly's offence in an identity mix-up.
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. Andreolli scrambled a dangerous cross by Bjorn Helge Riise to safety before keeper Doni had to make a sharp save from Jonathan Greening's shot.
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.
Given their dominance, it came as little surprise when Hodgson's side surged ahead in the 24th minute. 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 Mark Schwarzer.
Roma 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.
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 but Andreolli stole in unmarked at the far post and lashed home from six yards out with the last kick of the game.