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Muzzy Izzet went close in the ninth minute as he stretched to meet a Billy McKinlay cross but from eight yards out he drove his header over the top.
Matt Elliott blundered minutes later, his poor clearance falling to midfielder Alan Quinn. He fired in a low, hard cross and striker Leon Knight instinctively stuck out a foot and stabbed the ball wide.
Adams' side almost snatched the lead in the 23rd minute, another McKinlay cross causing problems in the Wednesday back four. Defender Frank Sinclair was first to react with a scooping shot, but the ball flashed over the top from ten yards.
Knight was foiled by Ian Walker minutes later as the on-loan Chelsea striker ran onto a Gerald Sibon pass. The 19-year-old weaved past a defender and struck a stinging 20-yard shot that was pushed round the foot of the near post by the Leicester keeper.
Sinclair twice went close with downward headers, the ball slipping over the Wednesday bar, as Leicester picked up the pace.
Lucky was firmly with Wednesday in the 68th minute after Paul Dickov carved out space on the right flank.
He picked out Callum Davidson lurking on the edge of the box. The Leicester full back drilled in a low shot and the ball hit the head of Jamie Scowcroft and looped over the stranded Kevin Pressman, hitting the underside of the bar before landing in the keeper's arms as he lay sprawled on the goalline.
Substitute Lloyd Owusu, on for the injured Knight, almost found a way through after a blunder in the Leicester rearguard but his firm strike from the edge of the area was safely gathered by Walker