Welbeck, an ex-Arsenal striker, and Julio Enciso both missed opening-period opportunities as Brighton took control of the game early on.
As Brighton went to the fourth round, a rotated Arsenal team tried to put more pressure on the game but was unable to do so.
In their penultimate Premier League match before the World Cup on Saturday, Arsenal will want to bounce back against Wolverhampton Wanderers, while Brighton will visit Aston Villa on Sunday.
Welbeck, an ex-Arsenal striker, and Julio Enciso both missed opening-period opportunities as Brighton took control of the game early on.
Reiss Nelson picked up the ball inside his own half, drove towards the Brighton defence, and found Nketiah in the 20th minute to give the home team the lead after withstanding that wave of Seagulls pressure.
After scoring five EFL Cup goals in five games last season, the Englishman peeled off onto the left side of the box and floated a lavish effort beyond Jason Steele's grasp into the far corner.
However, Arsenal's happiness would not last for long as Hein hauled down Welbeck in the area just five minutes after Nketiah scored the first goal after sliding while attempting to cut out a through play.
As Brighton went into halftime with a 1-1 score, Welbeck dutifully walked up to convert the penalty kick while refusing to celebrate against his former employers. However, early in the second half, it was all Arsenal.
As a result of Nketiah's long-range shot hitting the post and Steele's excellent stop on Nelson's goal-bound header in the 49th minute, Arsenal's inability to take advantage of both opportunities ultimately cost them the game.
After Jeremy Sarmiento saw Mitoma running down the left side of the field in the 58th minute, Brighton immediately burst forward. The Japanese attacker cut inside on his right foot before finding the far corner.
After Mitoma's goal, the energy of the hosts was depleted, and Brighton quickly scored three goals when Lamptey sprinted down the left flank, avoided Nelson's attention, and slid the ball home from a tight angle in the 71st minute.
As Brighton went to the fourth round, a rotated Arsenal team tried to put more pressure on the game but was unable to do so.
In their penultimate Premier League match before the World Cup on Saturday, Arsenal will want to bounce back against Wolverhampton Wanderers, while Brighton will visit Aston Villa on Sunday.
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