Lewis Dunk put the hosts ahead when he volleyed in a set-piece from Anthony Knockaert, who also hit the post during a dominant first-half display.
But Ross Wallace, who scored in the Owls' 2-0 first-leg win, saw his cross drift in for the equaliser that secured a 3-1 aggregate win.
Wednesday, out of the top flight since 2000, will now face Derby or Hull.
The Tigers take a 3-0 lead into their home second leg of the semi-final on Tuesday.
No team had previously overturned a two-goal first-leg deficit to reach the final of the second-tier play-offs - and Brighton made every attempt to create a bit of history.
Boosted by the availability of both Knockaert and Dunk after injury and suspension respectively, the Seagulls made a lightning start and could easily have been in front in the tie by the time Dunk netted from close range.
Knockaert struck the woodwork with a curling free-kick and also headed wide from five yards before crossing for Dunk to halve Brighton's deficit.
Any momentum from that opening goal quickly disappeared, however, as Wallace's cross beat everybody in the penalty area and nestled in David Stockdale's far corner.
The goal was allowed to stand, despite protests by Dunk that he had been pushed by Gary Hooper as the ball was played into the box.
Wednesday, undoubtedly second best in the first period, were rarely threatened after half-time as Brighton fell to a third Championship play-off semi-final defeat in four seasons.
Wednesday - who finished sixth in the Championship table, 15 points behind Brighton - are one win away from returning to the top flight for the first time in 16 years.
Since their relegation in 2000, the Owls have spent four years in League One and had only registered two top-half finishes in the second tier before this season.
Dejphon Chansiri targeted Premier League football within two years when his consortium took control of the the South Yorkshire club in January 2015.
And, having supported head coach Carlos Carvalhal heavily in the transfer market this season, the Thai businessman could get his wish ahead of schedule.
Sheffield Wednesday boss Carlos Carvalhal: "It's been a long time for Sheffield Wednesday and it's very important for the club.  Now we can wake up the giant.
"Let me give a word to Brighton and their manager. They had a fantastic season, they fought until the last minute to get promotion. They deserve promotion.
"But that doesn't mean we don't deserve it. We have grown up together like a team."
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Brighton manager Chris Hughton: "It's always difficult in the second leg to be 2-0 down, even at home, but we had great belief we could turn it around.
"We should have been at least one up before we got the goal. But things have gone against us in the last few weeks, and their goal was a foul on Dunk.
"That makes it tougher - at 1-1 they have motivation. That was really frustrating. The disappointment is massive."
Chris Sutton, ex-Blackburn striker on BBC Radio 5 live
"I really feel for Brighton, but credit to Sheffield Wednesday for their second-half performance.
"They were extremely comfortable. They had the strokes of luck with Brighton missing chances and a couple off the line in the first half.
"Over the two legs, Brighton haven't had the luck and it's Sheffield Wednesday who are celebrating.
"Glenn Loovens and Tom Lees were terrific at centre-half for Sheffield Wednesday, Keiren Westwood was commanding in goal and they are one game away from the top flight."
Match ends, Brighton and Hove Albion 1, Sheffield Wednesday 1.
Second Half ends, Brighton and Hove Albion 1, Sheffield Wednesday 1.
Attempt missed. James Wilson (Brighton and Hove Albion) left footed shot from outside the box misses to the right. Assisted by Kazenga Lua Lua.
Foul by Lewis Dunk (Brighton and Hove Albion).
Atdhe Nuhiu (Sheffield Wednesday) wins a free kick in the defensive half.
Attempt missed. Fernando Forestieri (Sheffield Wednesday) right footed shot from more than 35 yards misses to the right following a fast break.
Attempt blocked. Daniel Pudil (Sheffield Wednesday) left footed shot from more than 35 yards is blocked. Assisted by Fernando Forestieri.
Corner,  Brighton and Hove Albion. Conceded by Jeremy Helan.
Attempt saved. Lewis Dunk (Brighton and Hove Albion) right footed shot from the right side of the box is saved in the centre of the goal.
Attempt blocked. Liam Rosenior (Brighton and Hove Albion) right footed shot from outside the box is blocked.
Foul by Anthony Knockaert (Brighton and Hove Albion).
Jack Hunt (Sheffield Wednesday) wins a free kick in the defensive half.
Attempt missed. Anthony Knockaert (Brighton and Hove Albion) left footed shot from the left side of the six yard box misses to the left. Assisted by Lewis Dunk with a headed pass.
Attempt missed. Kazenga Lua Lua (Brighton and Hove Albion) right footed shot from outside the box is close, but misses to the left. Assisted by Lewis Dunk.
Foul by Bruno (Brighton and Hove Albion).
Glenn Loovens (Sheffield Wednesday) wins a free kick in the defensive half.
Corner,  Brighton and Hove Albion. Conceded by Tom Lees.
Attempt missed. Atdhe Nuhiu (Sheffield Wednesday) right footed shot from outside the box misses to the left. Assisted by Barry Bannan.
Atdhe Nuhiu (Sheffield Wednesday) is shown the yellow card for a bad foul.
Beram Kayal (Brighton and Hove Albion) wins a free kick in the defensive half.
Foul by Atdhe Nuhiu (Sheffield Wednesday).
Attempt missed. James Wilson (Brighton and Hove Albion) left footed shot from the centre of the box is high and wide to the left. Assisted by Beram Kayal.
Foul by Jiri Skalak (Brighton and Hove Albion).
Fernando Forestieri (Sheffield Wednesday) wins a free kick on the left wing.
Bruno (Brighton and Hove Albion) is shown the yellow card for a bad foul.
Foul by Bruno (Brighton and Hove Albion).
Jeremy Helan (Sheffield Wednesday) wins a free kick in the attacking half.
Corner,  Brighton and Hove Albion. Conceded by Sam Hutchinson.
Kazenga Lua Lua (Brighton and Hove Albion) wins a free kick on the left wing.
Foul by Kieran Lee (Sheffield Wednesday).
Corner,  Sheffield Wednesday. Conceded by Steve Sidwell.
Foul by Liam Rosenior (Brighton and Hove Albion).
Fernando Forestieri (Sheffield Wednesday) wins a free kick in the defensive half.
Liam Rosenior (Brighton and Hove Albion) wins a free kick in the attacking half.
Foul by Fernando Forestieri (Sheffield Wednesday).
Substitution, Brighton and Hove Albion. Liam Rosenior replaces Gordon Greer.
Attempt missed. Atdhe Nuhiu (Sheffield Wednesday) header from the centre of the box is too high. Assisted by Kieran Lee with a cross.
Offside, Brighton and Hove Albion. Jiri Skalak tries a through ball, but Steve Sidwell is caught offside.
Attempt blocked. Kazenga Lua Lua (Brighton and Hove Albion) right footed shot from outside the box is blocked.
Jiri Skalak (Brighton and Hove Albion) wins a free kick in the attacking half.
Sheffield Wednesday held off a spirited Brighton side to reach the Championship play-off final at Wembley on 28 May.