Hamilton, a summer signing from step three Alvechurch, converted sharply from a Jason Oswell nod down on just 14 minutes.
Bucks boss Gavin Cowan will be pleased with the workout in significant heat, as his side completed a second friendly without any obvious injuries, with important minutes into the legs of the players.
But Drayton, who play two levels below, gave an impressive account of themselves and went close to what would have been a deserved equaliser on a couple of occasions. It was a different contest to that won at a canter 6-1 by Telford last September.
Telford edged 1874 Northwich 2-1 in their first friendly - and first action since last season was curtailed in January - on Tuesday night.
Cowan named an unnamed trialist in the middle of his back three. Centre-half is a position the club are looking to add to and it is understood the trialist could finalise a move to the Bucks.
Also handed a start was new forward Hamilton, who was a successful trialist in the club’s early trial friendlies, and switched from lower-ranked Alvechurch.
Arlen Birch, who recently rejoined the club from Chorley, began at right wing-back.
Youth team player Jez Abbey, younger brother of new signing from Wolves Jed, also started in an attacking role behind the two strikers.
Drayton, under manager Richard Brown who is heading into his first season in charge, lined up in a 4-1-4-1 shape. Ash Rawlins started between the sticks, having returned to Greenfields from Telford this summer.
An even, well-contested start saw step four Drayton give as good as they got, including a couple of forays into the Bucks’ box in the opening seconds.
The visitors began to settle, with Henry Cowans busy in midfield and the lightning-quick Hamilton making sharp darts up front.
Oswell dragged just wide across goal shortly after 10 minutes in the game’s first opening of note.
But moments later Telford edged ahead. A looped Birch cross from the right was nodded down by powerful ex-Wrexham man Oswell for Hamilton, who turned sharply on the six-yard line and finished well into the bottom corner.
The National League North visitors could have extended their lead in the 15 minutes that followed, but Oswell first sent a header wide at the near post before a very presentable header, from another Birch cross, flew over.
Drayton, who lost the corresponding friendly against Telford 6-1 last summer, competed well and enjoyed some pressure five minutes before the break. Striker Reece Jacobs sent a decent chance well over from the corner of the box.
Cowan made five alterations at the break, introducing Adam Walker, Dom McHale, Jed Abbey, Ross White and a trialist, who played at left wing-back.
But it was hosts Drayton who started the sharper, somehow failing to convert a couple of low crosses into the six yard box, with the Telford backline stretched.
The second half was a calmer affair in terms of chances in the final third, and Northern Premier League First Division West Drayton continued to impress with their work and use of the ball. The flow was not helped by further changes to both around the hour.
Bucks player-coach Matthew Barnes-Homer almost doubled the lead midway through the second half after a smart turn but was denied by Rawlins, who also kept out the speedy Harry Bower moments later.
Trialist Devon Kelly-Evans’ error went unpunished with just over 10 minutes left as a Drayton midfielder fired over from distance.
The hosts continued to press and the lively Dimitri Dunkley shot over from the edge of the box after a good run when he had options to square.
Cowan’s men really should have made the game safe eight minutes from time as Adam Walker, who scored the winner at Northwich in midweek, fired a sitter over unmarked from 12 yards after more good work from Bower, who also fired narrowly over in the closing stages.
Market Drayton Town (4-1-4-1):
Rawlins; Riley-Stewart, G Taylor, Hughes, Price; Taylor, Dunkley, Burroughs, Russo, Davies; Jacobs.
AFC Telford United (3-5-2):
Griffiths (Scott, 63); Streete (White, 45), Trialist (McHale, 45), Lilly (Trialist, 72), Birch (Kelly-Evans, 63); Daniels (Trialist, 45), Cowans (Walker, 45), Byrne, Jez Abbey (Jed Abbey, 45), Oswell (Bower, 63), Hamilton (Barnes-Homer, 63).