City, who missed a penalty in first-half stoppage time with the score 1-1, have now not won in their last eight league games.
The Seagulls, winners of only one of their previous seven home league games, made the perfect start when Portuguese midfielder Joao Teixeira gave them the lead after only eight minutes.
Cutting in from the left, Teixeira drove in a low, right-footed shot which went into the net underneath the body of keeper Darren Randolph.
Brighton could only hold onto their advantage for five minutes as striker Clayton Donaldson hit his first goal in seven games by steering in a shot from the edge of the area by Stephen Gleeson.
Albion manager Chris Hughton gave full home debuts to former Celtic midfielder Beram Kayal and striker Leon Best, who is on loan from Blackburn, and Best should have hit the target mid-way through the half when he blazed over following good work by Rohan Ince.
The Seagulls had a let off shortly after as Andrew Shinnie shot wastefully over with the goal at his mercy after Stockdale failed to hold on cleanly to the ball.
Good work from Teixeira, on loan from Liverpool, helped set up Calderon on the edge of the box and his right-foot shot flew narrowly wide of Randoph’s right-hand post.
Stockdale came to Brighton’s rescue at the other end by blocking a goal-bound shot from Donaldson following a strong run through the middle by Lloyd Dyer.
Birmingham wasted a glorious opportunity to go ahead in first half stoppage time when Spanish defender Bruno Saltor brought down Shinnie inside the box, but Stockdale dived full length to his left to keep out Paul Caddis’ penalty.
The Seagulls regained the lead five minutes after the break through Calderon, who steered the ball passed Randolph on the edge of the six yard box following an early ball in by Sam Baldock.
Teixeira struck again to make it 3-1 on 58 minutes from close range after keeper Randolph could only parry a goal-bound effort from Baldock.
Birmingham boss Gary Rowett went three up front by introducing strikers Wesley Thomas and Lee Novak to join Donaldson just after the hour mark.
But Brighton continued to threaten and Teixeira was off target with a shot which took a deflection.
Defender Jonathan Grounds fired a rare opening for Birmingham over after substitute Thomas laid the ball to him.
Thomas gave the Blues late hope with his sixth goal of the season, turning neatly inside the area to beat Stockdale with a low shot after Brighton had given the ball away.
But Calderon made it 4-2 five minutes from time with a shot from close in after Lewis Dunk headed on a free kick from Dale Stephens. Substitute Novak pulled another goal back for Birmingham in stoppage time with a low shot.