Mo Diame's spectacular 72nd minute goal broke the Owls' hearts at Wembley as the Tigers edged the all-Yorkshire affair.
Defeats means Wednesday missed out on the £170million cash windfall which promotion would have brought.
As expected, head coach Carlos Carvalhal freshened up his midfield, bringing in Sam Hutchinson for his first start in over two months at the expense of Alex Lopez. Hutchinson has missed a big proportion of the campaign due to a bout of glandular fever.
It was Hull who had the first sight at goal, with Tom Huddlestone seeing his rasping 20-yard drive kept out by Keiren Westwood at the second time of asking.
With so much riding on the showpiece fixture, it was perhaps understandable that things were fairly scrappy in the early skirmishes.
Tigers centre-back Michael Dawson became the first player to be booked after nine minutes when he hacked down Fernando Forestieri. It was a hefty challenge but Wednesday's top-scorer maintained his composure and poise, forcing Eldin Jakupovic into action from the resulting free-kick.
Driven on by 40,000 Wednesdayites, Carvalhal's side stayed on the front foot and Ross Wallace's long range free-kick was pushed to safety by Jakupovic, who was deputising for the injured Allan McGregor.
Hull, who finished nine points better off than the Owls in the regular Championship season, almost made the breakthrough just before the half hour mark through Abel Hernandez. The Uruguay international's header was hacked away off the line by Kieran Lee. From the loose ball, the Tigers had a penalty appeal waved away when Dawson's effort struck Lee.
Hernandez, recruited for a club-record £10m nearly two years ago, continued to cause problems and Westwood produced a stunning save to prevent him from registering after Tom Lees failed to cut out the through ball.
Hull were enjoying a good spell of pressure, with full-back Moses Odubajo's low strike flashing narrowly wide.
Moments later, Mo Diame had his head in his hands after smacking the near post from seven yards out after gliding past three Owls defenders. It was another big let off for Carvalhal's side.
Westwood was by far and away the busier goalkeeper and the Republic of Ireland international superbly denied Odubajo before the interval after the Hull defender beat the off-side trap.
Wednesday's fans were in great voice, pushing their team on throughout a tight contest.
It took an inch-perfect last-ditch tackle from Curtis Davies to prevent Forestieri from breaking clean through on goal after the re-start.
Bu the Owls continued to live dangerously and Westwood had to be alert to deny Dawson at the back post.
Hull, relegated from the top-flight last season, should have taken the lead through Andrew Robertson following a rapid counter attack. Robertson some how blazed over when it looked easier to score after a fine cross by Odubajo.
Hutchinson, deployed just in front of the back four, ventured forwards on the hour mark, drilling a right-foot shot over the top after a strong run.
After Diame's exquisite strike, Carvalhal took off the injured Hutchinson, throwing on Atdhe Nuhiu to beef up their forward line in the closing stages.
Wednesday struggled to break down a well-organised Hull outfit, with Bannan firing over from distance.
As the Owls piled more men forward, it left themselves short defensively at times and skipper Loovens superbly thwarted Hernandez after another dangerous Tigers raid.
Hull: Jakupovic; Odubajo, Dawson, Davies, Robertson; Elmohamady, Huddlestone, Livermore, Snodgrass (Clucas 81); Diame (Maguire 88), Hernandez (Meyler 85). Substitutes: Kuciak, Bruce, Maloney, Akpom.
Wednesday: Westwood; Hunt, Loovens, Lees, Pudil (Joao 86); Wallace (Helan 63), Lee, Hutchinson (Nuhiu 76), Bannan; Hooper, Forestieri. Substitutes: Wildsmith, Matias, Lopez, Sasso.
Referee: Bobby Madley (West Yorkshire)