Having been the better team for the first hour, Welling had to dig deep to claim a point.
Although twice ahead, Weston's strong finish meant that they had to settle for a point.
New signing Tom Richards replaced Christian Nanetti in an otherwise unchanged Welling line up.
Richards was the first player to have an attempt but his effort from a half cleared corner flew wide.
Bradley Ash then earned his way into the referee's note book for a ludicrous dive as Chris Lewington came out to claim a long all forwards.
Lloyd Humphries shot well wide for Weston as Welling only half cleared a free kick before Sam Hatton tried to catch Luke Purnell out with a free kick that the Weston stopper claimed.
Matt Fish hit an effort for Welling but Ben Gerring half blocked it and it ballooned over.
Adam Coombes then headed over from Dan Walker's cross.
Walker was again in business down the right wing soon after when he cut in and sent in a low shot that Purnell saved at his near post.
Ashley Kington shot over for the Seagulls then Jacob Cane did likewise after catching Jordan Brown dithering.
Back at the other end Walker headed across to Coombes but his effort went wide of the far post with Purnell floundering.
Once again, a half cleared Weston corner fell nicely for one of their players to chance his arm but once again, Humphries' effort from outside the area was well off target.
An excellent deep Richards cross picked out Walker at the far post but he should probably have done better with a volley at the far post that failed to hit the target.
That could be forgotten moments later, when Coombes put Welling ahead in first half injury time. Jason Pope's attempted header back to his goalkeeper was anticipated by the Wings' leading scorer and he gave Purnell no chance with a crisp finish from the edge of the box.
After the break, Welling continued to play the better football and Coombes again tested Purnell with a drive that the keeper kept out at full stretch.
Scott Bartlett, Weston's manager, made a double change and it paid immediate dividends. Welling conceded a needless free kick and Dayle Grubb stepped up and gave Lewington no chance with his team's first effort on target.
Within a minute, Welling regained the lead when Danny Waldren burst into the Weston box and fired home.
On 78 minutes, Weston levelled when Grubb worked his way across the edge of the area before firing home.
That gave the hosts the momentum and only Lewington stood between them and an unlikely victory.
First he saved from Ash and then from a header from Jake Lee. Grubb, searching for his hat-trick was next to test Lewington but again the Welling keeper was equal to it.
At the other end, Coombes had a shot saved by Purnell but the last chance, deep into injury time, fell to Ash but he thrashed his effort high and wide.
Welling United: Lewington, Fish, Brown, Hatton, Hayles, Francis, Walker (Williams 90), Waldren, Crawford, Coombes, Richards (Slabber 79). Subs not used: Leacock, Fuseini, Johnson.