Aaron Kuhl, back following his three match suspension due to a red card way back in January, marked his return with his first goal for the club as early as the sixth minute.
Great work from Joe Lawal led to George Hunt crossing in, and Kuhl drove his finish past goalkeeper Mahoney from inside the box.
Welling went close to a quick leveller when Krabbendam did well down the left and whipped in a dangerous ball, but the header was off target.
The hosts had the better of the play and Bradley Stevenson had Jonathan North scurrying across his goal on 22 minutes, but the striker’s shot was safely past the post.
Slough’s best chance to extend their lead came minutes later when Tom Derry headed narrowly wide from a Guy Hollis long throw.
Eitel Goueth was lively down the left side for Slough, and a strong run led to the ball falling to Hunt, whose shot was blocked.
Stevenson dragged a shot agonisingly wide on 43 minutes after he was found in space down the left, but he had better luck a few minutes later in stoppage time when he drove in a shot from the edge of the box into the bottom corner.
Early in the second period, a powerful header from Pearce at the back post flashed wide of the near post.
But on 54 minutes Welling took the lead. A long ball through was latched on to by Ben Allen, who got to it before a committed North. He was able to take the ball around the goalkeeper and slot home into an unguarded net.
Just three minutes later, Goueth missed a great chance to level the scores when he blazed the ball over the bar from the edge of the box.
Goueth was again at the heart of the action on 68 minutes as he went on a mazy run which ended in him getting a shot away, but Mahoney saved.
Despite the increasing pressure from the Rebels, the hosts could have added a third goal as Dipo Akinyemi bore down on goal, but North made a good save at his near post.
However, just as Slough’s hope was fading away, Bird rose highest to meet a cross from the left and plant a header past Mahoney to bring the scores level.
That was not the end of the drama though, as substitute Ben Harris got a header on target deep into stoppage time but was unable to beat Mahoney.
Welling United Line up
Mahoney, Barnes, Pearce, Shokunbi, Durojaiye, Akinyemi, Perkins, Krabbendam, Pepe, Stevenson, Allen. Substitutes: Ezennolim, Saunders-Henry, Campbell, Hinds, Brown.