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Welling gave Luton a FA Trophy scare before bowing out in stoppage time on Tuesday.
Just 404 fans braved the cold at Park View Road - the lowest for a competitive Luton match in their history.
It looked as though the Wings were going to cause an upset but two goals in the last eight minutes saw the full-time team progress.
After Sunday's goalless draw, Welling player/manager Jamie Day (pictured) and midfielder Joe Healy returned from suspension at the expense of injured duo Andy Pugh and Andy Sambrook while Luton once again fielded a fringe line-up.
It took nearly half an hour for the first shot on target and Welling had only an off-target Loick Pires shot to show for their efforts when they took the lead at the end of the first half.
Visiting keeper and skipper Mark Tyler made a hash of Jack Oberstellar's high cross under pressure from Luis Cumbers and the ball somehow ended up in the back of the net.
Welling were in control early in the second half but Tyler's excellent save to deny Cumbers a second goal proved decisive as the hosts tired.
Luton levelled the scores on 82 minutes when Jason Walker fired in from the edge of the box.
With extra-time looming, Luton won the game in the third minute of stoppage time through Alex Lawless' unstoppable drive after a corner was half-cleared.