But Grove left it late for there was barely a minute to go when substitute Steve Brannan drove home the equaliser.
Leek appeared to be on course for victory when they established a two-goal lead early in the second half.
In a rather scrappy opening, Jimmy McCarthy had a shot tipped over and also saw another effort parried away by Steve Hodgson, while Ross Davidson went closest for Kidsgrove with a cross-shot from the edge of the box.
Leek opened the scoring on 41 minutes when a long ball over the top of the Grove defence sent McCarthy clear. His cross into the six-yard box was converted by Sean Turner.
Leek doubled their lead on 52 minutes following a corner by McCarthy. The ball was headed away by Wayne Johnson, but only as far as Christian Millar, whose half-volley went in off the keeper's body.
Dave Walker blasted high over for Grove, who suffered another setback when they lost the injured Davidson.
Meanwhile, Leek boss NeilBaker withdrew striker Matt Haddrell in a tactical move.
Kidsgrove found a way back into the game after 76 minutes when Danny Roberts flapped at a corner kick from Brannan and Walker was on hand to touch home Richard Smith's cross.
Walker, Mark Beeston and Tom Urwin all spurned chances for the home side, but they finally levelled on 89 minutes.
A Danny Griggs corner was cleared by Darren Tinson to the edge of the Leek box from where the lurking Brannan drove his shot across the keeper and into the far corner of the visitors' goal.