More on KentOnline
Welling were beaten by a controversial goal in National League South on Tuesday night.
Referee Jack Packman over-ruled his assistant and ignored an offside flag to award Chelmsford a corner nine minutes from time.
But Welling then only had themselves to blame as Alfie Pavey gave away a needless penalty and Lee Barnard settled the game from the spot.
After their 3-2 defeat at Truro on the opening day of the season, Welling drafted in keeper Dillon Barnes on loan from Colchester to replace Chris Lewington, who has left the club.
Toby Ajala has also departed Welling after just one game - he was subbed at half-time against Truro - with Sean Francis coming into the starting XI in a re-jigged defensive line-up.
it didn't take long for the game to spring into life when Welling player/manager Jamie Coyle made an early clearance. His challenge left Chris Dickson grounded and - after seven Chelmsford players rushed to referee Jack Packman - the official decided not to produce any cards.
Welling settled quickest and a stunning pass from midfield by the impressive Connor Dymond sent Bradley Goldberg clear. Chelmsford keeper Sam Beasant did well to parry his low shot and the Wings striker was unable to turn in the loose ball.
Former Wings defender Chris Bush was booked for a handball right on the edge of the penalty area after 18 minutes. But Christian Nanetti's low free-kick hit the defensive wall.
Chelmsford's Chris Dickson shot well wide midway through the first half before Kudus Oyenuga headed over from Michael West's free-kick five minutes before the break.
Referee Paxman, who was in danger of losing control throughout the first half, had another big decision to make on the stroke of half-time after deciding that Bush's late challenge on keeper Barnes was a foul. However, rather than produce a second yellow card, the official decided to give the defender a verbal warning alongside visiting skipper Elliot Omuzus.
The game became more direct in the second half as Chelmsford got on top. Welling were unable to maintain possession while they did not get tight enough to close the visitors down in midfield.
Michael West fired over and Oyenuga had an effort deflected wide for a corner.
A fine block by Ian Gayle denied Matthew Johnson with 16 minutes to left.
Welling started to lose their discipline, as they were unable to get out of their own half. Jack Parkinson was cautioned for dissent when the referee awarded the controversial corner and then Pavey's needless challenge gifted Chelmsford the points.
Oyenuga nearly added a second late on but Chelmsford were never threatened as Welling struggled to create anything going forward.
Welling: Barnes, Gayle, Jeffers, Parkinson, Coyle, Francis, Dymond, Healy, Bradbrook (Pavey 75mins), Goldberg, Nanetti (Joseph-Dubois 63mins). Subs not used: Dubojaiye, Pattison, Brown.
Chelmsford: Beasant, Omuzus, Braham-Barrett, Davies, Bush, Spillane, Johnson, Church, Oyenuga, Dickson (Barnard 63mins), West (Giles 75mins). Subs not used: Haines, Stevenson, Daley.
Attendance: 611.