Ryan Giggs and Wayne Rooney scored in the space of two second-half minutes and United could have had more but for the superb goalkeeping of Manuel Neuer with the final score hardly reflecting the huge difference in quality between the sides.
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Sir Alex Ferguson's team were superior in all aspects of the game and United - three-time winners of Europe's top prize - go into the return leg having gained a huge advantage after dominating the dazed Germans for about an hour.
"It was a top performance and credit to the players for their belief in themselves and the trust in each other," Ferguson told reporters. "It ranks as one of our best (performances in Europe)."
Whether attacking from the right, left or centre, the home side's defence looked sluggish with United's players having time and space to create numerous chances.
Neuer twice denied Giggs late in the first half but the Welshman, picked out in the box when Rooney flicked a pass through Joel Matip's legs, was on target in the 67th minute.
"I think there was frustration towards the end of the first half with that final save from Ryan's (header). We had to correct that at half-time," said Ferguson.
"But what broke the camel's back was Chicharito's goal (that was disallowed early in the second half) because we saw that we could beat the guy (Neuer)."
"This is strange because his peak seems to have lasted so long now," said Ferguson. "I see no evidence of this waning."
"He won't be playing on Sunday (against Arsenal), he'll be playing next Wednesday (in the Schalke return) and when he gets that freshness he shows no signs of weakness. He is an amazing man."
Ferguson even hinted that he might be able to rest some players for the second leg at Old Trafford, saying: "Depending on the result on Sunday I could make two or three changes".
"We had imagined this game differently. We lacked the final pass today which we had against Inter Milan (in the quarter-finals)," said Schalke coach Ralf Rangnick.
"We had to thank Manuel Neuer that we were not two or three goals down in the first half.
"But why should we capitulate? We need to make things differently and in football many things have happened.
"We have shown we can score away from home and we will try to get our chances in Manches
