The Saints were just too hot to handle for Gavin Cowan's side, racing into a three-goal lead thanks to James Armson, Lee Ndlovu and Carl Baker.
Marcus Dinanga came up with a response for the Bucks, but they did not have the legs to mount a fightback.
They remain fifth in National League North but are now only two points in front of eighth-placed Boston United.
Last Tuesday's FA Trophy heroics proved to take too much out of Telford as they slumped to defeat at Brackley.
The Bucks entered this clash at St James' Park against a fellow play-off contender in exceptionally high spirits after a 2-1 last-16 win at Spennymoor Town.
But they were desperately short of energy.
The 400-mile round trip to County Durham, and a gruelling tie on a notoriously boggy Brewery Field pitch, was too much to overcome.
Cowan is not one to make excuses, but those miles on the clock cannot be overlooked.
Brackley continuously found pockets of space and were first to the vast majority of loose balls.
Credit to them too, they converted their opportunities with aplomb.
You just hope Telford can find that second wind to pick up at least a point against Guiseley at the New Bucks Head on Tuesday, and this just ends up going down as a bad day at the office - not a defeat which knocked Cowan's charges off course.
Telford made four changes from the side which beat Spennymoor.
Top scorer Daniel Udoh and last month's marquee signing Dinanga returned up top, while England Under-19 international Josef Bursik was back between the sticks and Brendon Daniels on the left again.
Harsh on Andre Brown, Amari Morgan-Smith, Henry Cowans and Andy Wycherley - but you could argue this was the strongest XI available to Cowan.
The possible exception to that would be defensive rock Theo Streete, who was on the bench.
So, in that respect, it was surprising to see the Bucks begin the clash in such a sluggish manner.
By the 23rd minute, they were deservedly a goal down as Armson swept home Matt Lowe's cross from the right.
Telford spent a lot of the first period in their own half, especially struggling to contain Baker and Lowe.
Ndlovu doubled Brackley's lead with a close-range header from another Lowe cross, and Baker coolly rounded Bursik to slot in the third.
But Dinanga struck on the counter-attack just before the interval for the Bucks, perhaps giving them a lifeline.
You have to suspect it was a fiery half-time team-talk from Cowan, and a change was made for the start of the second half as Streete replaced Daniels.
Darryl Knights soon took the place of James McQuilkin too. Telford could not find that needed increase in tempo though.
The Saints were happy to sit back, and the Bucks, who also brought on Brown for Steph Morley, were not able to break them down.
Brackley just had more in the tank on a disappointing day for Telford.
23 - Goal! Brackley take the lead - and it has been coming. Lowe fires the ball across the face of goal and Armson is there to poke it home from close range. Clever finish.
40 - Goal! The hosts are two to the good. Lowe puts in another cross from the right flank, this time floating it to the far post where Ndlovu nods past Bursik. Too easy for the Saints.
42 - Goal! Brackley swiftly find a third. Baker is not tracked and stays onside, latching onto a through ball and skipping past Bursik before putting the ball into the net. Well taken.
44 - GOAL! Telford pull one back! Against the run of play, Dinanga drills into the corner with his left foot after controlling a ball over the top of the Saints defence. Maybe there is a chance of a comeback.
Brackley (4-5-1): Lewis; Myles, Hall, Dean (c), G Walker; Lowe (Nti, 90), Byrne, A Walker, Armson, Baker (Fairlamb, 89); Ndlovu
Subs not used: Stirling-James, Jackson, Jeffers
Goals: Armson (23), Ndlovu(40), Baker (42)
Telford (3-5-2): Bursik; White, Sutton (c), Morley (Brown, 80); Barnett, Royle, Deeney, McQuilkin (Knights, 56), Daniels (Streete, 46); Udoh, Dinanga
Subs not used: Cowans, Morgan-Smith
Goals: Dinanga (44)
Referee: Lewis Smith (Wigan)