Mourinho's side revived after the break but Sir Alex Ferguson's champions should have come home with at least two away goals, such were the number of chances that fell their way.
When Ferguson first began guiding United in the European Cup in the mid-Nineties, they were regularly punished, particularly before crowing Italian crowds, for laxness in possession. Not now. Ferguson has taught his players how to keep the ball, a strength that served them so well here.
Carrick embodied this expertise at ball-winning and distribution in front of the watching England manager, Fabio Capello. Mourinho must have been impressed too.
Following Mourinho's pantomime-cum-sermon on the eve of the game, 80,000 had thronged excitedly into this famous arena, anticipating drama of the highest order.
In keeping with the theatrical theme, "Edinburgh Festival Reds'' were present, clearly resting before their August acting duties. Football's pre-gentrification generation were more fully represented, typified by a banner that read: "United, Kids, Wife – In That Order''.
Surprise rippled through these United fans on hearing that Wayne Rooney was being kept in reserve. Yet Ferguson has pulled off such shocks before, notably with a similar midfield against Chelsea and any Doubting Thomases were swiftly silenced. Gaffer's cracking up? No chance. Ferguson knew what he was doing, sending his players out in 4-2-3-1 formation, clearly wanting to protect a back-four missing Nemanja Vidic.
How they responded, dominating the first half with a wonderful display of possession football. When Patrice Evra and Jonny Evans wasted the ball just before the interval, gasps of disbelief almost sprung from the terraces. And how United actually reached the break without a goal was a travesty of sporting justice, pure and simple. Inter became so disheartened in the first period that their reserve keeper, Francesco Toldo, was even cautioned for dissent.
Every man in red responded to Ferguson's instructions and his call to arms. Darren Fletcher, impressively focused after the distressing news of his partner being robbed at knife-point, joined Carrick in shielding the defence while Giggs, Cristiano Ronaldo and Park Ji-sung pushed on in support of Dimitar Berbatov.
Ronaldo clearly relished the task, the European Footballer of the Year possibly stung by all Mourinho's claims that Zlatan Ibrahimovic was in his league. So inspired, Ronaldo could have laid claim to the match-ball within 10 minutes. He twice went close with free-kicks, and in between demanded a desperate save from Julio Cesar. Running on to a Giggs corner, Ronaldo thundered in a header that Inter's keeper somehow clawed away.
United were composed in possession, working triangles around black-and-blue shirts, thrilling their fans. Ferguson clearly ordered his players to dictate the tempo, dominating possession, taking the sting out of the crowd. Inter's fans had been so pumped up for the European champions' visit that even United's kit-man, Albert Morgan, was treated to some choice Italian invective before kick-off.
In seeking to silence the Inter faithful, Carrick and Giggs were immense, rarely risking losing the ball, always threatening something special. Carrick dribbled through the middle, a sinewy run of growing menace, his enterprise curtailed only by a trip from Javier Zanetti. The Argentinian even lectured the Englishman on diving. The referee, Luis Medina Cantalejo, was having none of Zanetti's appeals, signalling a free-kick for United. Up stepped Giggs, up went the free-kick, but again failing to beat Cesar.
Inter had pursued Giggs' signature all through the Nineties, and their players tried vainfully to catch him here. Racing on to a sublime Carrick pass, the Welshman could have squared to Park and Berbatov but instead went alone, gliding past Rivas Lopez. Frustratingly for United, Giggs' shot was brilliantly saved at the near-post by Cesar, whose excellence prevented humiliation for Mourinho's side.
Still United dominated. Cesar saved another Ronaldo dead-ball special. Then the World Player of the Year glanced a header just wide, again following a fine cross from the ubiquitous Giggs. United were rampant and Inter were relieved to make the break intact.
Emerging from the tunnel, Ferguson predicted to television reporters that there "will be a reaction from Inter'' and he was right. Mourinho had lifted his players' mood, and strengthened his defence by removing the hapless Rivas Lopez.
Here were the Inter who led Serie A. Here was Ibrahimovic showing some class. When Esteban Cambiasso crossed, Ibrahimovic let the ball run on to Adriano, whose shot flew wide. The warning was clear: Inter were waking.
The Italians screamed for a penalty when Ferdinand challenged Adriano, who dissolved on impact like a melon in a blender. Contact was minimal and Medina Cantalejo again dismissed Italian appeals.
United needed to be careful. Ibrahimovic was moving with more conviction, pooping up on both flanks, particularly United's right where O'Shea at times looked vulnerable. United needed Ronaldo to show, to help take the game to Inter.
The champions were under pressure, Ferdinand being bullied at times by Adriano. Fortunately for United, some of their number were giving their performances of the season. Evra and Carrick made vital interceptions. But United were defending too deep, inviting Inter on. And so the chant went up, for "Rooney, Rooney'', for their lionhearted striker to cause Inter some concerns.
United still threatened on the counter, and Park was a whisker away from turning in Ronaldo's cross.
The goal that Giggs deserved almost arrived with 15 minutes left, United's captain dribbling through only for his shot to be blocked. Yet the best chance fell to Cambiasso, who missed from close range.
Rooney then arrived – with a bang, the forward clattering into Cambiasso with a force that could have brought red, rather than the yellow brandished by Medina Cantelejo. The second leg promises fireworks.