Alan Lee gave them an early advantage, which Leon Clarke levelled to ensure the score was 1-1 at the break.
Darren Ambrose put the London side ahead after 62 minutes, leaving Wednesday an hour to score the two goals needed to survive.
They managed just one. Darren Purse scored after 87 minutes to set up a tense finale, but Wednesday could not find a way through for a winner.
Champions Newcastle United ended the season with all three points thanks to a 1-0 win at Queens Park Rangers, with Peter Lovenkrands scoring the goal.
West Bromwich Albion, already confirmed as runners-up, drew 1-1 with Barnsley. Graham Dorrans, the subject of a transfer bid from Premier League side West Ham United this week, levelled Hugo Colace's opening goal.
Nottingham Forest finish the season third after their 2-2 draw at Glanford Park. They will take on Blackpool in their play-off semi-final, after the Seasiders secured sixth place with a 1-1 draw at home to Bristol City.
Swansea City could have snatched sixth place with a victory at Doncaster Rovers, but they were held to a goalless draw. Lee Trundle had a late goal disallowed for the Welsh side.
There will be one Welsh side in contention for a place in next year's Premier League, although Cardiff City prepare for their game on the back of a 2-0 loss at Derby County. Jay McEveley and David Martin scored for the Rams.
Their opponents Leicester City, in contrast, have won their last five matches. Yaan Kermorgant and Richie Wellens scored in a 2-0 win over Middlesbrough.
Watford spent all season searching for an away win, and finally recorded one on the final day of the campaign.
They did it in some style, recording a 4-0 success at Coventry City thanks to two goals from Danny Graham as well as strikes from Henri Lansbury and Marvin Sordell.
Sheffield United were comfortable 3-0 winners at Ipswich Town, while Reading were 4-1 home victors over Preston North End.
Plymouth Argyle and Peterborough United will meet again in League One next season. Both sides had their relegation confirmed long before today's game, with Posh registering a 2-1 success at Home Park to end the season on a positive note.