United, promoted last season from League One, stunned the home crowd by going 2-0 up within a quarter of an hour thanks to John Fleck's sweetly-struck low drive from an indirect freekick in the third minute and then Clarke's cool finish after he was left one-on-one with Wednesday keeper Keiren Westwood.
Wednesday, who lost out in the promotion playoffs in the past two seasons, pulled back a goal on the stroke of half-time when striker Gary Hooper hooked in the ball at the near post and after the break they piled on the pressure.
That effort paid off when Portuguese substitute Lucas Joao equalised with a well-taken left-foot strike in the 65th minute but United responded superbly.
Within a minute of the restart, winger Mark Duffy made a fool of Wednesday's Dutch defender Joost van Aken and then fired home from a tight angle to make it 3-2.
Clarke then made sure of the win, latching on to a clever pass from the outstanding David Brooks, holding off two defenders and then slotting home.
United's Sheffield-born manager Chris Wilder was delighted with his team's display after they scored four goals at Wednesday for the first time in the 124-year history of the fixture.
"We have had to wait a long time as a football club and we have had some ups and downs. We stepped out and played some outstanding football. I thought we were outstanding in every department," said the former United defender.
(Reporting by Simon Evans; Editing by Clare Fallon)
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