Carlos Carvalhal's side will be at Wembley on May 28 to face Hull City or Derby County, with the former leading 3-0 ahead of Tuesday's home return leg, for a place in the top flight which could be worth up to 170 million pounds ($244.73 million).
Brighton took the lead through a Lewis Dunk volley after 19 minutes but a controversial Ross Wallace equaliser just before the half-hour mark, after Dunk seemed to be pushed in the area, ensured the game finished level with the Owls in the final.
“We didn't play our usual game and I think the players were a bit nervous,” said Carvalhal, whose sixth-placed side upset an Albion team that finished third. “After what Brighton did this season they don't deserve this moment but that is football.”
“To go to Wembley and play is fantastic. We will be 50-50 in the final but we will fight and do our best,” he added.
No team had overturned a semi-final, first-leg deficit of two goals or more in the second-tier playoffs but Brighton were determined to try, with Anthony Knockaert hitting the post with a free-kick after 10 minutes as they piled on the pressure.
Keiren Westwood kept Brighton out with some fine saves but nine minutes later the hosts went ahead on the night as centre-back Dunk volleyed in with his right foot at the back post from Knockaert's free kick after defender Tom Lees missed the ball.
However, in the 28th minute Wallace crossed from the right and the ball evaded everyone and flew into the far corner with the referee ignoring appeals for a foul on Dunk who appeared to be pushed by Gary Hooper when challenging for the ball.
Wednesday had finished sixth in the table to take the last playoff spot, 15 points behind Chris Hughton's Brighton, but it is the South Yorkshire side who are one game from returning to the top flight for the first time since 2000.
It was Brighton's third Championship play-off semi-final defeat in four years as their bid to return to the top flight for the first time since the 1982-83 season continues.
Original source: Sheffield Wednesday into Championship playoff final