Welbeck clipped in a clever finish in the 114th minute at St James' Park as the Seagulls recovered from falling down to Alexander Isak's early penalty.
A bright start to proceedings yielded half-chances at both ends, but Newcastle were given a golden opportunity from the penalty spot after Tino Livramento was fouled clumsily by Yankuba Minteh.
Isak - back starting after a groin injury - made no mistake from 12 yards as he thundered his kick beyond the reach of Bart Verbruggen.
Isak thought he had his second of the afternoon 10 minutes later after powering a snapshot in at the near post but was denied, having strayed just offside.
But shortly before half-time, Minteh - formerly on Newcastle's books - made amends for his earlier error by surging through to equalise with Brighton's first shot on target.
The game's aggro rose in the second period when Anthony Gordon was sent off after elbowing Jan Paul van Hecke in the head before Brighton's Tariq Lamptey was dismissed for a cynical foul.
Fabian Schar thought he won the game late on for Newcastle with a controlled finish, however it was scrubbed off with the new semi-automated offside technology showing the tightest of margins.
Brighton upped the ante in extra time and were rewarded six minutes from time through Welbeck, who broke the offside trap to lift a shot high into the net over Martin Dubravka.
Later on Sunday, reigning FA Cup champions Manchester United host Fulham at Old Trafford.