Coming into the game with Wealdstone, Welling United had completed five straight home league wins.
That run was ended when the Stones made the most of Welling's absence of a cutting edge and former Wing Ricky Wellard made them pay.
Although Welling had the lion's share of the goal attempts, most were from outside the penalty area and they paid the price for failing to take any of them.
Danny Waldren, for Welling, was the first to try his luck when picking up a knock out but his first time effort flew wide.
Edward Oshodi then made room for himself on the edge of Welling's penalty area but he was also unable to find the target.
Elliott Godfrey fired over from distance, as did Aryan Tajbakhsh before Welling tested Jonathan North for the first time. Sam Hatton's free kick was heading for the top corner before North intervened with a spectacular save.
Chris Lewington struggled with a Wealdstone cross and it fell for Danny Green but he, too, was unable to find the target.
Rickie Hayles flicked wide from a Hatton free kick as Welling started to come into the match a little more and both Karagiannis and Waldren had shots blocked.
Waldren then hit a powerful drive from 25 yards the North did brilliantly to tip away at full stretch.
Karagiannis put a shot into the trees behind the goal at the park end and Green shot well wide at the other.
As half time approached, another Karagiannis effort was deflected wide for a corner from which Welling came close to going ahead. Hatton's delivery picked out Hayles, but his firm effort flew back off the bar.
Tajbakhsh shot over after being set up by Wellard but Welling were soon trying their luck again and Matt Fish's long shot dipped just after clearing the bar.
Finally, after 62 minutes, Wealdstone scored the crucial goal. Tom Hamblin put in a delicate cross to the far post and Wellard stooped to head home from close range.
Moments later, Welling may have levelled when Rhys Murrell-Williamson crossed but Harry Crawford was unable to apply the finish.
They then missed an even better chance when Murrell-Williamson again crossed well but Karagiannis sliced his shot wide when he looked certain to score.
Green had a shot on target, but not with sufficient power to trouble Lewington.
Substitute Jamie Slabber headed wide from another Murrell-Williamson cross before having another header saved comfortably.
Karagiannis was blocked twice in one attack before firing over in the next one.
North saved comfortably from a Waldren free kick before Lewington ran the length of the pitch for an injury time corner only to head wide.
Welling United: Lewington, Fish, Brown, Hatton (Slabber 69), Hayles, Francis, Karagiannis Waldren, Walker, Crawford, Murrell-Williamson. Subs not used: Agu, Leacock, Richards, Fuseini.