Derby, which plays in the League Championship, ended a 14-game unbeaten streak for the English, European and world champions. This is Derby's first major cup semifinal since 1976.
Man United did not start many of its top players, bringing in Cristiano Ronaldo and Wayne Rooney as substitutes. The second leg is Jan. 20 at Old Trafford.
Commons gathered a pass from Mile Sterjovski and lashed a 25-yard shot past goalkeeper Tomasz Kuszczak.
"It will be tough to go to Old Trafford and hold on to the 1-0," Commons said. "We'll enjoy it this week - just to beat Manchester United and be able to say that to your kids and grandkids is going to be very special."
Derby was managed for one game only by David Lowe before Nigel Clough takes over Thursday. Clough is the son of famed Derby manager Brian Clough, who guided the club to a championship in 1972. He was introduced to the Pride Park crowd before the game.
Nigel Clough is now 90 minutes away from reaching the final of the competition his father won four times with Nottingham Forest.
Tottenham was favored to be the opponent in the March 1 final after trouncing Burnley 4-1 in Tuesday's first leg.
