Blues opened the scoring against the run of play early in the opening half and then had to rely on Ben Chapman to keep them in it midway through the half.
But Chapman could do little to maintain the lead when the visitors were awarded a penalty two minutes from half time.
There were very few chances in the second half with both sides seemingly cancelling each other out but Market Drayton stole the points with a strike just five minutes from time.
Chapman took his place in goal after arriving on loan from Nantwich Town to take the place of injury victim Danny Roberts whilst manager Lee Caswell made three changes to the side that lost at Sheffield in the previous game in Aden Johnson, Ben Nixon and Wayne Corden.
Drayton were the first to show when they came close to opening the scoring in the 4th minute. A poor defensive ball forced Chapman to race to the edge of his box to try and head clear but the ball fell to Tom Roger and he looped a shot wide of the far post.
Three minutes later the visitors wasted three opportunities to get in a shot as they worked their way along the 18-yard line before Glyn Mellor blocked Stuart Ellis’s shot.
However, Blues took the lead in the 11th minute when James McCarthy evaded the offside flag to advance on goal and plant a shot wide of the keeper into the far bottom corner.
In the 23rd minute Drayton mounted an attack when Nicky Porter turned inside to get in a superb shot that saw Chapman make a fantastic save pushing the ball over his crossbar.
A minute later Chapman launched a long clearance deep into the Drayton half and when Matt Haddrell headed on to McCarthy he nipped in to volley on the turn but the effort was wide of the target.
In the 31st minute, Tim Sanders played a good ball down the right channel for McCarthy who created a chance with his first time cross on the run that saw Haddrell go full length only to power his header narrowly over the crossbar.
Two minutes from half time a Drayton free kick was only half cleared and when the ball was pumped back into the box the referee spotted pushing and awarded a penalty. Drayton substitute Martyn Davies claimed the ball and sent Chapman the wrong way to level the scores.
The second half was largely uninspiring with chances mainly coming from set pieces.
Drayton keeper Andy Pryce tipped a Mellor header from a free kick wide of the upright and then Darren Tinson headed wide from the corner.
In the 76th minute McCarthy raced clear to advance into the box but his shot was deflected by the keeper and cleared by a defender.
Five minutes from time Drayton wrapped up the points when the Leek defence backed off and a cross into the box found Roger in space and he neatly lobbed the ball over Chapman into the back of the net.
Blues had a chance to level the scores deep into added time when Haddrell set up Tim Grice
but he sliced wide from a good position.
Leek: Chapman, Sanders, Johnson, Tinson, Mellor, Ward (Woolliscroft 80), McCarthy, Nixon, Haddrell, Millar, Corden. Sanders (Grice 80). Subs not used: Bridge, Holmes, Edwards.
Market Drayton: Pryce, Johnson (Haynes 90+2), Peel, Hartlebury, McMullen, Blake, N Porter (Davies 37), S Porter, Reynolds (Stevens 79), Roger, Ellis. Subs not used: Cohen, Stallard.
Report: Mike Cope