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Welling scored a stoppage-time winner from the penalty spot to reach the National League South play-off final.
Brendan Kiernan held his nerve to score a spot-kick five minutes into additional time after the Wings had blown a 2-0 lead in Sunday's semi-final.
Welling will now visit Woking in the final next Sunday, where the winner will be promoted to the National League.
After a quiet opening, Welling started to get on top with Danny Mills heading over Bradley Goldberg's right-wing cross.
They took the lead on 27 minutes. Gavin McCallum's free-kick caused all sorts of problems in the visiting defence and as Chelmsford's Adrian Cascaval tried to clear, the ball bounced off Goldberg and went past keeper Nathan McDonald.
Five minutes later, Goldberg twisted one way and then the next before McDonald could only parry out his 20-yard drive and the ball was hacked behind for a corner.
When the corner was delivered, Welling doubled their lead. Thierry Audel's acrobatic attempt came back off the inside of the upright and hit the back of McDonald's leg before trickling in for an own goal.
Chelmsford got themselves back into the game a minute before half-time when Tom Wraight headed in Billy Knott's free-kick.
Chances after the break were few and far between, Mills heading over for Welling one of few efforts at goal for the home side.
Chelmsford never forced Welling keeper Dan Wilks into serious action until the 83rd minute when substitute Jonny Giles let fly from 25 yards. Wilks appeared to have turned the ball over the bar but instead he ended up looping it only into his own net.
The game seemed destined to go into extra-time as Josh Hill headed wide in injury time for the Wings but there was one more twist.
Former Dartford man Chez Isaac tripped Kiernan as he worked his way into the penalty area and the referee was in a great position to award the spot-kick.
With Goldberg substituted earlier, it was left to Kiernan convert the penalty and send Welling through to the final.
Welling: Wilks, Gibbons, Braham-Barrett, Mendy, Acheampong (Hill 46mins), Audel, KIernan, Ijaha, Mills, Goldberg (Bettamer 74mins), McCallum. Subs not used: Hajizadeh, L'Ghoul, Annau.
Chelmsford: McDonald, Omozusi, Anderson (Reynolds 53mins), Spillane, Cascaval, Whelpdale, Porter (Isaac 62mins), Knott, Wraight, Murphy, Fenwick (Giles 68mins). Subs not used: Church, Barnum-Bobb.
Attendance: 1,912.