Colin Gordon's side looked on course to have relegation confirmed when they trailed to Kadell Daniel's early strike with just eight minutes remaining.
But Kelvin Langmead headed in an 82nd-minute equaliser from close range.
And, just three minutes later, James McQuilkin converted Ben Whitfield's cross to leave Harriers 11 points from safety with four matches to play.
Harriers remain bottom, three points behind 23rd-placed Welling, who have played a game more.
Kidderminster Harriers beat Welling as two late goals kept their slim hopes of National League survival alive.