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Skipper Lee Clarke came to his side's rescue with a second-half header to see off Brislington in the FA Cup on Saturday.
Clarke nodded in from close range after keeper Chris Humphries made a mess of punching clear Jack Obersteller's 68th-minute free-kick.
It is the first time that Welling had reached the first round of the FA Cup for eight years, when they were knocked out by Huddersfield in 2005.
But they made hard work of defeating a side that play four levels below them in the non-league pyramid, in the Western League.
Welling started brightly enough and forced a number of early corners. Jake Gallagher helped one cross on at the near post but it was cleared to safety while Blaine Hudson fired over from the edge of the box minutes later.
But there was little action of note thereafter as Welling's delivery into the penalty area, in particular, let them down.
Clarke set up Loick Pires early in the second half but, despite getting clear of the last defender and rounding keeper Humphries he was forced too wide and the chance was gone.
It took Welling 54 minutes to have a shot on target when Gallagher's low drive was saved while Brislington nearly took advantage of a Hudson slip at the other end but Omari Simms shot wide.
Once ahead, Welling continued to have the advantage possession-wise but the home side raised their game and Tom Godsell volleyed straight at keeper Lee Butcher with four minutes left.
There were seven additional minutes played, as play was stopped twice for flares and then a smoke bomb as the occasion got to the home fans, while a fan also ran onto the pitch to familiarise himself with Butcher's water bottle.
In the end, however, there were no further scares for Welling and Jamie Day, who reached the first round for the first time in his managerial career.