Ricky Hayles gave the visitors an undeserved lead as he capitalised on a defensive error to head home a corner on 26 minutes.
Chris Dickson equalised for the Clarets just seconds into the second half. He took advantage of comical defending from the visitors to smash the rebound home.
Spillane put City ahead with a delightful diving header three minutes later, before Shaun Jeffers sealed the victory with a beautiful curling effort with two minutes left.
The Clarets had a strong start to the game with Louie Theophanous coming closest to putting City ahead. The physical striker controlled well on the edge of the box before striking a strong shot at Chris Lewington, however the Wings’ shot-stopper got a strong palm behind the ball to deny the home side.
Dickson almost gave Chelmsford an eleventh minute lead after Josh Hill passed inside, however the veteran striker could only watch his shot curl agonisingly wide of the post.
The visitors arguably should have been awarded a spot-kick moments later. Alex Cathline danced his way between Michael Spillane and Chris Bush before falling to the ground, however Whistler Josh Smith waved away the Wings’ striker’s claims.
City had a goal disallowed on 20 minutes when Theophanous struck Dickson’s chest down beyond Lewington, however the official had his flag raised for offside.
But the home side were left to rue their missed chances, however, as Ricky Hayles gave the visitors the lead. A Welling corner was swung into the box, Ross Fitzsimons came to punch the cross but missed, leaving Hayles an open goal to head home.
The Clarets came bursting out for the second half and got themselves in-front.
Dickson levelled with just a mere matter of seconds gone in the second half. The City striker took full advantage of a defensive error, won the ball from Lewington and nestled the ball in the bottom corner.
And moments later the Clarets were ahead. Theophanous’ aimed cross was deflected wide off Dickson’s header, but Spillane used a fantastic diving header to give the hosts a lead from Willmott’s corner.
The hosts could have doubled their advantage moments later but Dickson saw his shot crash off the post, before Willmott saw his clever cross evade Dickson at the far post.
Welling United changed their shape on the hour-mark, with the visitors replacing wingers Dan Walker and Rhy Murrell-Williamson with Harry Crawford and Dean Leacock.
Rod Stringer made his own change a few minutes later as he replaced Elliott Buchanan with Shaun Jeffers.
Chances came at a premium for either side, but City could not increase their lead as Dickson and Theophanous both came close again.
City finally sealed the victory two minutes from time. Substitute Jeffers picked the ball up on the edge of the box and curled a delightful shot into the top corner of the goal to seal City’s place in the play-offs.
1. Ross Fitzsimons
2. Josh Hill
3. Luke Daley
4. Max Cornhill
5. Chris Bush © (90', off for Mark Haines)
6. Michael Spillane
7. Robbie Willmott
8. Josh Rees
9. Louie Theophanous (81’, off for Kane Haysman)
10. Chris Dickson
11. Elliott Buchanan (64’ off for Shaun Jeffers)
12. Kane Haysman
14. Shaun Jeffers
15. Ashley Miller
16. Daniel Blanchett
18. Mark Haines
1. Chris Lewington
2. Matt Fish
3. Jordan Brown
4. Sam Hatton
5. Ricky Hayles
6. Phil Walsh
7. Dan Walker (59’, off for Dean Leacock)
8. Danny Waldren
9. Alex Cathline (81’, off for Pierre Joseph-Debois)
10. Tyrone Marsh
11. Rhys Murrell-Williamson (59’, off for Harry Crawford)
12 Ali Fuseini
14. Pierre Joseph-Dubois
15. Harry Crawford
16. Tom Richards
17. Dean Leacock
Chelmsford City: Josh Rees (43’), Chris Bush (45’)
Welling United: None.
Chelmsford City: Chris Dickson (46’), Michael Spillane (49’), Shaun Jeffers (88’).
Welling United: Ricky Hayles (26’)
Half Time Score: Chelmsford City 0-1 Welling United
Full Time Score: Chelmsford City 3-1 Welling United
Man of the Match: Chris Dickson