Town trailed at the break after a defensive lapse left Tyrese Hall unmarked at the back post to turn home from a cross and leave Swindon with work to do in the second half.
But after the break, Danny Butterworth and Sean McGurk both scored fabulous goals to see the game turn on its head.
Three days after progressing in the FA Cup, Swindon make it beyond the group stages in the BSM Trophy for the first time since 2021.
Ian Holloway made 11 changes from his first line-up in charge as he looked to give all of his players a chance to impress. Academy defender Antony McCormick was handed his first senior appearance alongside Ryan Delaney, who played at The Nigel Eady County Ground as a Swindon player for the first time.
As Holloway predicted before the game, Spurs came out of the gate playing the Ange Postecoglu way, keeping possession nicely and looking to work it quickly into advanced areas. Town were having to sit in a bit more and try and take the sting out of the game when they could.
But they were creating openings of their own in transition and had a very good one as Jake Cain clipped a cross from the left corner of the box right onto the boot of Botan Ameen. The Iraq youth international lashed a volley back across goal and narrowly over the top.
The skill of the Spurs youngsters was evident when Leo Black picked up the ball on the left touchline after 17 minutes. The midfielder took the ball and briefly became Diego Maradona as he weaved through defenders like an ambulance in traffic but when it came to the shot it was soft and too close to Dan Barden.
The academy side had more of the ball but it was blow-for-blow with chances. Cain made a great turn in midfield to get behind Tottenham before spraying the ball out to McGurk. He was able to set Rosaire Longelo away to clip the ball over Sam Archer but the defenders got back to clear.
After 25 minutes that quality showed again as Yusuf Akhamrich rolled away from a tackle and played an inch-perfect cross to the far post for Hall to turn home from the edge of the six-yard box. Questions would be asked about how Swindon left him so free.
The goal appeared to spook Swindon and they got away from what was giving them success. McGurk and Butterworth had been picking up positions in between the lines to get away from Tottenham but instead, they were trying to get Paul Glatzel and Ameen in behind from deep too quickly and giving the ball away.
This led to a long period without chances until the 41st minute when Longelo’s cross was headed to the top of the box. Cain came onto it, beat a man, and let fly with an effort that skimmed off the turf and went just wide of the post.
Holloway came into the game saying he wanted to see how his team would interpret his instructions, and they would need to be re-established at the break if Swindon were going to get the result they needed to progress.
The task almost got bigger straight away in the second half as Tyrese Hall got in behind substitute Grant Hall down the left and played a ball across the face of goal. Damola Ajayi got on the end of it but Barden got down with lightning reactions to keep the deficit to one.
Seconds later the Spurs goalkeeper pulled off an even better save when McGurk got the ball on the edge of the box. He unleashed a laser towards the top corner, but the young stopper flicked it with the very top of his fingertips against the crossbar.
From the resulting corner, Town put one under the posts and had two players unmarked at the back post but again Archer was there to flick it over them.
But the chances kept coming as Butterworth played a quick one-two with Glatzel to get in down the side of the box and he slotted his effort calmly into the back of the net.
They had to wait a bit longer for it but when the winner came it was an absolute delight. In the 74th minute, Glatzel took down a cross in the middle and spread the play to McGurk. He took a touch and with no back lift curled one over the goalkeeper and into the top corner.
Two wins from two in the cup competitions for Ian Holloway gives him a good base to go into the Milton Keynes Dons games and a few selection headaches to boot.
STFC starting XI: Barden, Delaney, Cotterill, Glatzel, McGurk, Longelo, Cain, King, Butterworth, Ameen, McCormick.
STFC substitutes: Evans, Sobowale, Drinan, Hall, McGregor, Mitchell, Foye.
TH starting XI: Archer, Ashcroft, Robson, King, Cassanova, Kyerematen, Akhamrich, Black, Hall, Ajayi, Williams.
TH substitutes: Linton, Andyapen, McKnight, Irow, Maguire, Arganese-McDermott, Baptiste.
Attendance: 1,816.