United coach Alex Ferguson fielded five youngsters, while including Wayne Rooney in his starting line-up after recovering from injury.
He was partnered by French striker Louis Saha, who was given a rare start by the Scottish tactician.
Young defender Gerard Pique stunned the home crowd in the 34th minute when he rose unmarked to head home Nani's left-wing corner.
Brazilian winger Mancini drew Roma level in the 71st minute with a superb curling shot from inside the area.
Already-qualified United finished atop of Group F with 16 points, five clear of Roma, who were also guaranteed a place among the Champions League last 16.
Elsewhere, Sporting Lisbon, who will play in the UEFA Cup, crushed Dynamo Kiev 3-0 at home.
Defender Anderson Polga broke the deadlock from the penalty spot in the 35th minute.
Joao Moutinho doubled the lead seven minutes past the hour mark before Liedson completed the rout two minutes from time.