Brentford failed to make the most of large spells of possession and slack defending cost them the chance of a trip to Cardiff for the opportunity to make the leap into the Championship.
But Wednesday fans can now start dreaming of pushing their way back up the league pyramid as they attempt to return to their former glories.
Brentford went into the game knowing that they would have to score at least once in order to stand a chance of reaching the final after John-Paul McGovern gave Wednesday a first-leg lead.
However, the visitors started with all guns blazing and could have scored three times in the opening ten minutes.
First Peacock ran through on goal on eight minutes and looked destined to tuck the ball past Stuart Nelson, only for a bad touch to let him down at the vital moment.
Moments later Glenn Whelan hit a fierce drive which was well saved by the Brentford keeper and then Nelson pulled off another good stop as Richard Wood tried to prod the ball home from close range.
The Bees managed to survive a frantic start and before long they began to exert pressure on Paul Sturrock's men.
Jay Tabb had a shot deflected wide of the post on 14 minutes and then Alex Bruce managed to block a Sam Sodje header on the goal-line.
After 25 minutes Deon Burton tested David Lucas with the shot of the game. The Brentford striker hit a rasping volley from a Stewart Talbot cross, but somehow the Owls goalkeeper tipped it wide.
A defensive lapse two minutes later cost Brentford dear. Paul Heckingbottom's cross from the right was met by the unmarked Peacock and he made up for his earlier miss with a header into the bottom corner.
Martin Allen's men headed in at half-time desperate for a morale-boosting team-talk, but after emerging for the second half they soon fell further behind.
Substitute Brunt's 20-yard free-kick took a wicked deflection off Chris Hargreaves in the Brentford wall and the ball diverted past Nelson into the net.
The Bees had a mountain to climb and threw on Marcus Gayle in a bid to try and grab a lifeline. Isaiah Rankin, Sodje and Talbot all came close as Brentford urged each other forward in the latter stages.
And with two minutes left Andy Frampton struck a consolation goal to give home supporters something to celebrate.
But it was too little too late as Sturrock's men held out for a win which gives them the chance to return to the Championship.