Andy Pugh, a former Wings` player, gave the Darts a great start with an opening goal inside 3 minutes.
And 10 minutes into the second half, Pugh punished his former club with a second.
Welling`s efforts were rewarded in the 77th minute when Callum Driver scored with a fine free-kick from 20 yards.
And they were level 3 minutes from the end when Elliot Bradbrook fouled brother Tom in the box. Jeffrey Monakana stepped up to slot home the spot-kick.
But there was to be a late twist as Alfie Pavey volleyed in a spectacular winner in stoppage time to win it for the Darts.
Second-placed Havant & Waterlooville were unable to build on their win at Eastbourne Borough on Saturday with another success as they drew 0-0 against battling second-bottom Bognor Regis Town at Westleigh Park.
Now the Hawks are level with Chelmsford City, Braintree Town and Hampton & Richmond Borough on forty points.
Braintree and Chelmsford battled out a 2-2 Cressing Road draw in front of a crowd of 1,465.
Michael Spillane gave the Clarets the lead after just 7 minutes.
But they led for less than 4 minutes as Phil Roberts pulled the Iron back on level terms.
However, 9 minutes into the second half, Kudus Oyenuga cut inside and was brought down in the box and he dusted himself down before converting the resulting penalty.
That looked like being the winner, until 2 minutes into stoppage time when Marcel Barrington smashed home Braintree`s leveller.
Hampton were also held to a draw – 1-1 at the Beveree by Wealdstone.
The Beavers got off to a great start when Charlie Wassmer put them in front inside the opening 6 minutes.
But a goal 6 minutes before half-time from the in-form Abobaker Eisa eventually proved enough to earn the Stones a share of the points.
Now just point behind in sixth are Truro City, who won 3-0 at Tatnam against third-bottom Poole Town.
It was goalless at the break, but the deadlock was broken on the hour-mark when Cody Cooke gave the White Tigers the lead.
Eight minutes later, Tyler Harvey added the visitors` second and the same player made the points safe with a third 11 minutes from time.
Currently holding the final play-off spot are Hemel Hempstead Town after they shared a 2-2 derby thriller at Clarence Park against the team immediately below them and level on points, St Albans City.
This was also goalless in the first half, but Hemel took the lead twice in the second period, firstly through Joe Howe after 63 minutes, then Spencer McCall on 80 minutes.
But Sam Merson twice equalised, both times within minutes of the Hemel goal.
And the game finished with a red card to City’s Zane Banton after a poor first touch left him stretching into the next tackle and the referee had no hesitation.
Three points further back in this amazingly tight division are East Thurrock United, but their game at the Aspect Arena against Concord Rangers was abandoned after the lights went out midway through the first half and although they briefly came on at half-time, another failure early in the second half saw the game called off.
Weston-super-Mare`s excellent campaign continues and they are now tenth and just a point behind East Thurrock after leap-frogging hosts Bath City in the table following a 2-0 success at Twerton Park.
This was yet another game that lacked goals in the opening 45 minutes.
It wasn`t until 15 minutes from time that the first arrived, and it came from the Seagulls` Gethyn Hill.
A Nick McCootie own goal killed off Bath`s challenge in the 83rd minute.
Whitehawk are looking doomed already, as Boxing Day`s 1-0 Enclosed Ground defeat to mid-table Eastbourne Borough leaves Steve King`s side nineteen points adrift of the safety-mark.
Manny Odubade`s 20th minute penalty, awarded after Hawks` keeper Seny Dieng floored Ian Simpemba in the box, eventually proved enough, although the Hawks put the visitors under a good deal of pressure in the second half.
Oxford City ground out a 1-0 victory at Evesham against Gloucester City, who are now fourth-bottom.
Rob Sinclair’s 16th minute strike eventually proved decisive in a match that was far from pretty.
Chippenham Town suffered a third successive defeat, this time at Bulpit Lane to a Hungerford Town outfit who are now level on points with the Bluebirds and moved a place above them into fifteenth in the table.
A Will Richards own goal gave the Crusaders a 12th minute advantage that Darren Mullings cancelled out after 33 minutes.
But 7 minutes into the second half, Mullings` game was over as he was sent-off with a straight red card.
And the ten-men were punished 12 minutes from time when Alex Wall popped up with Hungerford`s winner.