Alan Smith, Wayne Rooney and Patrice Evra also scored for United, while Daniele De Rossi scored for Roma.
"It was a fantastic performance. The play, (the) penetration (was) absolutely superb," United manager Alex Ferguson said as his team reached the last four for the first time since 2002.
"We can't wait for the semifinals now.''
Carrick scored in the 12th minute after Ronaldo won possession on the right wing, beat Mancini and set up Carrick for a looping shot into the top right corner.
"To start so well and carve up the chances and score the goals we did was a great feeling," Carrick said.
"To carry it on for the full game was the stuff of dreams."
Smith, making his first European start in more than a year, doubled
United's lead in the 17th.
Ferguson said: "He's deserved it more than anyone. He's been patient. He's trained like a beast."
Rooney slotted home from close range two minutes later off a pass from
Ryan Giggs, and Ronaldo scored his first European goal in the 44th after
cutting inside from the right, beating two defenders and scoring at the near post.
The Portugal winger added a second in the 49th. After Roma struggled
repeatedly to clear a corner, Giggs regained possession and picked out
Ronaldo at the back post.
Ronaldo then set up United's sixth, sending a crossfield pass to Gabriel
Heinze, who picked out Carrick in the 60th.
Roma finally penetrated the United net in the 69th when Daniele
De Rossi turned home Totti's cross, but Evra scored in the
81st with a weak shot at the near post.
Francesco Totti had come close in the sixth, but his shot was deflected by Wes Brown.
However, the violence which marred the first leg match between the two sides returned before the match as fans clashed outside the stadium.
Fans threw bottles and firecrackers as riot police backed up by officers
on horseback struggled to regain control.
In the first leg, which Roma won 2-1 last week in Roma, police fought with English fans inside the stadium.
The violence will no doubt lead to further investigations by UEFA.