The Whites remain unbeaten at home since a 1-0 reverse by fellow promotion chasers Burnley in September and are now three points clear of the third-placed Clarets.
The Owls stay four points off the play-off places in 10th.
Defeat marks a tough end to what has been an awkward week for the South Yorkshire side.
On Wednesday night owner Dejphon Chansiri, who was at Elland Road on Sunday as some Owls fans protested against him, talked at length at a fans' forum and revealed that he had not spoken to boss Danny Rohl this month.
Rohl then said he was "surprised" by the Thai businessman's comments and urged the club to come together, with defender Di'Shon Bernard admitting the atmosphere was "tense".
They got off to a terrible start as Leeds scored with their first attack. Joe Rothwell's clever pass bisected the Owls defence but keeper Beadle looked to have it under control only to fail to retain the ball and Tottenham loanee Solomon showed composure to score with two Wednesday defenders on the line.
Thereafter the away side did grow into the game and home defender Ethan Ampadu did brilliantly to slide in and stop Josh Windass getting a shot off after being played in on goal and Yan Valery hit a low shot into the side-netting.
It looked like that early goal would be enough for Leeds after a largely uneventful second period but Ramazani's fine shot made the last few minutes comfortable and Japan international Tanaka flicked home an impudent finish in the final moments.
Leeds are now unbeaten in their past 18 matches against teams from Yorkshire.
They welcome Norwich on Wednesday, while the Owls host Bristol City the same night.