Second-half goals from Ashley Young and Patrice Evra denied Stoke a place in the last four of the competition for the first time in 41 years.
Referee Mark Clattenburg called a halt to proceedings for just under 10 minutes as strong winds and a hail storm swept through the Britannia Stadium.
He said: "I thought we made a fair fist of it in the first half against all the elements and thought we were in decent shape.
"Our organisation was good, United weren't creating too many chances.
"But United, on the night, deserved to win, they just had a little bit more power and pace.
"We weren't able to ask enough questions of them in the second half."
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