After an indifferent start, Inter produced a performance of which Barcelona would be proud; pressing high up the pitch and hitting Barca on the break with pace and incisiveness when they won the ball back.
With Lionel Messi subdued, Inter cashed in on their supremacy in devastating style and the Italian champions now stand on the brink of a first appearance in the final since 1972.
Yet it all started in typical fashion for a Barcelona side used to having their own way, as Guardiola's men dominated the early stages at the San Siro.
They capitalised on a Milito miss after Victor Valdes had saved from Samuel Eto'o, the visitors surging up the other end as Maxwell was allowed to get to the byeline down the left and pull the ball back for Pedro to coolly roll home his 20th goal of the season.
Inter were seeing less of the ball than their opponents, but they were dogged in defence and threatening in the final third and after Milito bent a hugely presentable chance wide, they equalised on the half-hour mark.
As Eto'o crossed from the Inter right, the Barca defenders all got sucked into the middle, including right-back Daniel Alves - and when Milito turned and teed up Sneijder, the Dutchman had the freedom of the park to lash in an equaliser.
Barca, denied space with which to operate, were struggling to find anything like their usual fluency as Inter doubled up on Messi and denied the influential Xavi the time to spray passes around.
Sneijder keeps his cool to level matters for Inter
Just after the break Goran Pandev failed by inches to latch on to Milito's ball from the right, but the Italians did not take too much longer to get themselves ahead.
Messi lost the ball in his own half and Inter broke with pace, the ball finding its way to Milito on the right-hand side of the area. The Argentine took his time and picked out the marauding Maicon as the Brazilian took a touch and rammed past Valdes.
Barca, perhaps weary after their 14-hour coach trip to get to Milan, tried to respond, but Messi's shot and Sergio Busquets' header were both well saved by Julio Cesar in the Inter goal.
On the hour mark, Mourinho's men took firm control of the tie. They broke quickly again and after Sneijder had completely misdirected his header from an Eto'o cross from the right, Milito was on hand to nod in from point-blank range.
Having looked comfortable early on, now Barcelona were flustered and even Xavi was misplacing passes, much to the delight of the Nerazzurri faithful.
But Barca recovered their composure to lay siege to the Inter goal in the closing stages and they will wonder how they did not at least narrow the deficit going into next week's tie.
Messi's 30-yard free-kick forced Julio Cesar to parry, before Alves was denied what appeared to be a good penalty appeal when he fell under Sneijder's challenge in the area. To rub salt into the wounds, the Brazilian found himself cautioned for simulation.
As the clock ticked down Gerard Pique - switched from centre-back to a target man role as Barca pressed for a goal - and Pedro were both denied by Julio Cesar as Inter clung on to their formidable advantage ahead of a daunting trip to the Camp Nou.
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