After dominating the first half-hour they conceded a penalty and were relieved to see Daniele de Rossi fire his spot-kick over the bar.
Roma wasted several other chances to score before the outstanding Owen Hargreaves crossed for Tevez to head home the only goal of the evening.
Rio Ferdinand was a surprise inclusion in United's starting line-up after recovering from the injury he picked up at Middlesbrough on Sunday.
Wayne Rooney and Cristiano Ronaldo were on the bench while Ferguson opted to start with Mikael Silvestre who has been sidelined since September, and Gerard Pique.
United began the game extremely brightly and Philippe Mexes was forced to clear Hargreaves' dangerous early cross away from danger.
Ji-Sung Park dragged an effort off target before the South Korean's looping cross found Tevez who was unable to guide his header towards goal.
An unmarked Wes Brown then headed over the bar when he should at least have hit the target and Ryan Giggs threaded the ball through to Hargreaves who was denied by Doni's smart save.
Mirko Vucinic forced Edwin Van der Sar to make his first save of the evening before Tevez and Hargreaves combined to set up Giggs who was again denied by Doni.
Another good save by Doni kept Tevez's effort out after more excellent play down the right-hand flank by Hargreaves before Roma were handed a controversial penalty when Brown was deemed to have brought down Mancini.
However, de Rossi's penalty was dreadful, sailing way, way over the bar as United breathed a sigh of relief.
The miss seemed to revitalise the Serie A side though and they dominated possession for a spell without creating any clear-cut chances until United broke the deadlock in the 70th minute.
Hargreaves was once again the architect, crossing for Tevez to guide a diving header past Doni.
The match won, Gary Neville was brought on for the last ten minutes to a rousing reception from the home fans as he tasted first-team action for the first time in 13 months.
United will now play Barcelona in the semi-finals later this month after the Catalan giants completed their victory against Schalke.