After Brede Hangeland had headed the Cottagers into a first-half lead it looked as though Roy Hodgson's team would be able to hang on to the win even after Stephen Kelly had been sent off with 14 minutes left.
But with virtually the last kick of the match, David Pizarro's corner found Andreolli who controlled the ball with his chest before hammering a shot into the net.
The Serie A team were without the injured Francesco Totti but Hodgson's side were the better outfit for long spells and were unfortunate not to take all three points.
Roma had dispensed with the services of Luciano Spalletti earlier in the campaign, replacing him with Claudio Ranieri, but came into the game in 11th spot in Serie A and had lost to AC Milan at the weekend.
They had also failed to impress in Europe, beating CSKA Sofia but losing to FC Basel and after this result, face a battle to win through to the last 32.
Fulham's far from full-strength side edged the first half with Jonathan Greening having an early low drive saved by Doni before taking the lead when Hangeland nodded Bjorn Helge Riise's corner into the net.
When the visitors responded, Mark Schwarzer was forced to make a great stop to deny John Arne Riise an equaliser but the Cottagers thoroughly deserved their interval advantage.
It was a different game after the break though as Ranieri's side dominated. Daniele De Rossi had the ball in the net only to be ruled offside and Schwarzer did well to save Mirko Vucinic's drive.
With 14 minutes left, Roma were awarded a penalty when John Arne Riise was upended by Stephen Kelly, who was sent off, but only after the referee had initially shown a red card to Hangeland before Kelly explained it was him who had committed the foul.
Eventually Jeremy Munez took the spot-kick only to see his effort superbly turned away by the outstanding Schwarzer.
Despite their numerical disadvantage Fulham looked set to hang on until Andreolli's last-gasp intervention.