Mark Schwarzer and David de Gea pulled off fine stops in an outstanding first half which also included a power outage, but the England striker decided the tie
Both sides had great chances throughout but the England striker showed nerves of steel to calmly finish inside the box after a mistake from Philippe Senderos let him in on 79 minutes.
Hugo Rodallega broke the offside trap with ease after four minutes, but his cut-back from Ashkan Dejagah was well blocked in the center.
United will scarcely believe it didn't take the lead in the seventh minute. A corner came in which was bundled towards the net with only Schwarzer's desperate dive preventing a certain goal. Evra hit the bar seconds later with the goal gaping, then saw another shot expertly saved by the keeper.
It was David de Gea's turn to excel after 11 minutes, superbly tipping over a wonderful John Arne Riise volley from 30 yards.
Riise shaped to shoot again three minutes later but opted to turn and feed Bryan Ruiz, who saw his cute volley tupped onto the post
United then hit the post with the ball deflecting high from a corner on to Schwarzer's bar, with Antonio Valencia screwing the follow-up wide.
There were no chances until the 32nd minute, when United hit the woodwork for a third time with Rooney curling an effort onto Schwarzer's left-hand post from 20 yards.
The floodlights then went out with five minutes of the half to play to the delight of the 25,670 fans, and while the players came off, power was restored within five minutes. Play resumed with a corner, with Schwarzer collecting a Van Persie drive at the second attempt.
Both teams were a little more cautious from the restart after each manager presumably told their sides in no uncertain terms to tighten up at the back.
Chances were few and far between but Chris Baird heated things up with a stinging drive on the hour that De Gea did well to smother.
Van Persie broke the offside trap a minute later and teed up Nani on the byline, whose eight-yard shot was well blocked by Baird, while De Gea was perhaps fortunate his parried save was not turned in by Ruiz with 20 minutes remaining.
United had half-chances while Jonny Evans did well to head away from underneath his own crossbar as the pressure rose, while Rafael prevented a certain goal by heading away from a corner on his own goal-line, but United struck the decisive blow.
Senderos tried to block Van Persie's run but only succeeded in letting Rooney run onto a long ball, and the striker made no mistake by curling the ball around Hughes and into Schwarzer's net.
Van Persie cleared another header off the line and Chicharito missed a one-on-one in a brilliant ending, but United saw out the game.