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The Wings looked to be heading for all three points until substitute Dayle Southwell turned the ball home from close range after a long throw-in was flicked on five minutes into stoppage time.
It was harsh on the Wings who produced an impressive defensive display to limit Grimsby to few chances - just days after shipping three goals in eight minutes at League 2 Plymouth in the FA Cup.
They had been given great encouragement after taking an 11th-minute lead through midfielder Healy's fifth goal of the campaign.
Kurtis Guthrie exchanged passes with Ross Lafayette and his cross found Healy in the penalty area. Healy chested the ball, then flicked it with his left foot before volleying beyond keeper James McKeown with his right foot.
Grimsby enjoyed plenty of possession but failed to seriously test Wings keeper Lee Butcher.
After the break, Grimsby nearly levelled when a free-kick came back off the inside of the post while Welling could have doubled their lead as Alex Dyer and Ross Lafayette were denied in quick succession by McKeown.
Dyer, who was playing in his final game for the Wings before joining Swedish second division outfit Östersunds, was named man-of-the-match but he could only watch on as Southwell bundled the ball in to level the scores with what was effectively the last kick of the game.
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