The veteran midfielder was booked in the 26th minute for a needless foul on Christian Wilhelmsson and saw red nine minutes later for a clumsy tackle on Francesco Totti.
Faced with playing for 60 minutes with a man down, United produced a gritty performance and rode their luck on occasions to end the match still very much in the tie after Wayne Rooney grabbed a priceless away goal.
United started the match with plenty of injury concerns with Gary Neville, Mikael Silvestre and Nemanja Vidic all absent and the home side took advantage with a slick first-half display.
Roma took full advantage of the extra man and it was no surprise when they opening the scoring on the stroke of half-time.
Totti took a quick corner and squeezed the ball through to Mancini whose cut-back to the penalty spot was forced home by Rodrigo Taddei via a large deflection off Wes Brown which left Edwin Van der Sar wrong-footed.
After the interval, Cristiano Ronaldo rose to the occasion and was the spearhead of United's increasing attacking moves.
And on the hour-mark the fleet-footed winger broke through two tackles before sending Ole Gunnar Solskjaer free down the right flank.
The veteran striker looked up and clipped over a cross to the back post where Rooney's superb control on his chest took Daniele de Rossi out of the game before the England international slotted a superb shot beyond goalkeeper Doni.
Roma though refused to accept a draw and launched wave after wave of attacks against a tiring United defence.
And they were rewarded after 66 minutes when Mirko Vucinic was on hand to roll home the winner after Van der Sar could only fist a powerful drive from Mancini back to the waiting substitute.
Roma could have taken a huge stride towards ending the tie ahead of next week's second leg but in the final ten minutes wasted a string of chances before Totti drove inches wide of Van der Sar's upright.