United were let off the hook by Daniele de Rossi after wasting a series of opportunities to clinch their place in the semi-finals of the Champions League.
Roma goalkeeper Doni produced three quality saves to deny Owen Hargreaves, captain Ryan Giggs and Anderson from adding to their 2-0 advantage in the second leg of this quarter-final tie at Old Trafford.
But then on the half hour, with United in control of the game, they breathed a sigh of relief when de Rossi drove a penalty over the bar after Wes Brown had brought down Mancini inside the area.
But that was as much as Roma mustered before Carlos Tevez sealed a last-four tie with Barcelona with his 17th goal of the season in the 70th minute.