The result sees Sam Wilkinson's men slip to 17th in the league with the prospect of a trip to seventh-placed Bedworth Town looming next Saturday.
Callum Ward and James Blunden returned after injury but skipper Phil Watt was absent due to the head injury he received at Alvechurch last week.
Market Drayton had four clear-cut scoring chances all game and managed to convert one of them in the second half.
Their first golden opportunity came after 10 minutes when striker Ryan Seal broke through the Whites' defence only to see his weak, left-foot shot trickle wide.
The Whites struck back soon after when a three-man move involving Smith, Hornsey and McGovern almost gave them the lead.
United keeper Sam Vince was called into action after 20 minutes to make a flying save to his right as he punched clear a long-range effort from Sean Williams.
What followed was a catalogue of misses by the home side who totally dominated their opponents for the rest of the half but incredibly, failed to find the net.
Keeper Haydn Whitcombe and his fellow Drayton defenders produced heroics to keep out a rampant United.
The second half started in the same mould as the first as a Whites' header was cleared off the line. This was followed by close efforts from Toyne, Grimes and Cotton.
However, the visitors' defence held firm with a combination of excellent goalkeeping and just a hint of good fortune.
In the 72nd minute, and against the run of play, Market Drayton took the lead following a quick break from defence which saw Sean Williams' shot hit the upright and fall to Judd Ellis who finished clinically.
Despite shouts for penalties by the Whites, coupled with goal-line clearances and defensive heroics by the visitors, the score remained 1-0 which means a seventh defeat in eight games in all competitions for the Ashby Avenue outfit.
It was a game United should have won at a canter.
However, chances must be converted and no matter how hard they tried, the Whites' strike-force could not find the back of the Market Drayton defence.
Man of the Match: Scott Matthews
Peter Hayward
# | Team | Pl | Pts |
6. | Stamford | 42 | 71 |
7. | Leek Town | 42 | 68 |
8. | Lincoln United | 42 | 68 |
9. | Corby Town | 42 | 60 |
10. | Cleethorpes Town | 42 | 55 |
11. | Stocksbridge Park Steels | 42 | 53 |
12. | Peterborough Sports | 42 | 51 |