The Premier League outfit were three goals up at the break but a stunning effort from Miralem Pjanic from inside his own half and a Francesco Totti penalty set up a frantic finish
The England striker had netted a double in the 7-0 demolition of LA Galaxy in Louis van Gaal's first game as United boss in midweek and he was at it again as the Premier League side scored three goals in the space of 10 minutes at the end of the first half to stun the Serie A side.
United had looked vulnerable at the back with a three-man defence before Rooney opened the scoring with a sublime right-foot strike, and then set up Juan Mata for a superb second goal soon afterwards.
Rooney struck from the penalty spot to add the third - just as he did against the Galaxy - after Roma new boy Urby Emanuelson had upended Danny Welbeck in first-half stoppage time.
A packed crowd inside Sports Authority Field had been treated to two moments of sheer quality from Rooney and Mata for the opening goals, but there was even better to come after the break when Miralem Pjanic, once linked with United, pulled a goal back 15 minutes from time with an audacious lob from inside his own half, to leave United's substitute goalkeeper Ben Amos red-faced.
Francesco Totti pulled another goal back from the penalty spot with time running out, but United held on to make it two wins out of two for Van Gaal.
Roma had made a promising start as United looked uncertain with their new formation, playing three central defenders, and Mattia Destro almost caught them out when he raced onto a ball over the top and lobbed young goalkeeper Sam Johnstone, but the ball bounced down and over the crossbar.
Leandro Paredes then saw his free-kick deflected over the wall as United looked rattled and Alessandro Florenzi forced a save out of Johnstone with a left-footed strike from outside the penalty area after Destro had found space again.
It was United who took the lead with a goal out of nothing 10 minutes before the break, though, as Rooney was allowed too much time to line up a strike from 25 yards out that flashed into the far corner of the net.
The former Everton man then caught Roma out once again just over two minutes later when he picked out Mata with an incisive pass, the Spain midfielder finishing with a well-executed lob.
Rooney helped himself to a second goal on the stroke of half-time, sending the goalkeeper Lukasz Skorupski the wrong way from the spot after Emanuelson had brought down Welbeck.
Roma were left with a mountain to climb, but Pjanic had the crowd on their feet with a sensational strike 15 minutes from time as he spotted Amos off his line and lobbed him from inside his own half.
Leandro Castan hit a post late on and then veteran Totti pulled another goal back from the spot after Michael Keane had been penalised for handball, but United stood firm to secure the victory and open their Group A campaign with a win.